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1929

Organizing a solid analytic geometry Organizing a solid analytic geometry

Elizabeth Agnes Flood The University of Montana

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ORGANIZIÏÏG

a

SOIID AmiYTIC GSOIÆETRY

by

Elizabeth A. Flood

Presented in partia l fulfillment of the requirement for the degree of Master of Arts.

State University of Montana

1929

(Signed) ______Chairman lxa4. Com.

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SOLIP AHAIYTIC G20IŒTRY

Chapter I

The Point

1. Reotangular coordinates.— In plane analytic

geometry a point in the plane is determined by a pa ir

of numbers (x,y) which give i t s distances of d irec tions

from two lin es a t righ t angles to each o ther. S im ilarly

a point in space may be determined by three re a l numbers

(x ,y ,z ) which give i t s d istances of d irec tions from three

fixed mutually perpendicular planes. These planes divide

space into portions called octan ts. The in te rsec tio n of

the thhee planes is called the o rig in . Distances along

the X—Eixis and to the rig h t of the o rig in are considered

positive and to the le f t negative; distances along the

y—axis and to the front are considered positive and toward

the back negative; and distances along the z^-axis upward

are considered positive and downward negative. The rec

tangular coordinates x, y, z of a point are always w ritten

in the order (x ,y ,z ) . That i s , (2,3,4) is the point whose

X —coordinate is 2, whose y coordinate is 3, and whose z

coordinate is 4.

2. Location of a point by coordinates.— Let the


coordinate planes be ZY, YZ, ZX, and le t P be any

point in space* Throngh P draw PM

perpendicular to the XY plane, PE ^r f o . -jL

perpendicular to the YZ plane, and PL J

perpendicular to the ZZ plane* The

distances HP, MP, IP are the coordi

nates of P; that is , HP is the x—coordinate of P, MP is

the z—coordinate, and IP is the y—coordinate.

3* Octants*— We have already observed that the

three mutually perpendicular planes XY, YZ, ZX, divide

space into eight parts or octants* The octants are num

bered by the signs of the coordinates of a point as

given in the table.

Coordinates Octants

1st 2nd 3rd 4th 5th 6th 7th 8th

X + — — + + — — +

Exercises

1. Locate the following points: (0,2,3), (-1,-11,7),(0,0,-4>), (0,b,0), (a,0,0l*

2* In what octants is the x—coordinate of a point positive? the y—coordinate? the z—coordinate?

3* I f the X and y coordinate of a point are positive, in what octants may the point be?


4* P rojections*— The p ro jec tion of a point P upon

a l in e is the po in t in which a lin e through P perpendic

u la r to the l in e , meets the l in e . ^

In the figu re , the p ro jec tion of P^ 2upon the x—ax is is IT% and the p ro jec tio n

of Pg on the same ax is is Eg,

The p ro jec tio n of a point on ^ plane is the foo t of

the perpendicular from the poin t P to the p lane. In the

above^ fig u re i s the p ro jec tio n of the point P^ on the

xy^plaine, and Mg is the p ro jec tio n of the po in t Pg on the

xy—plane•

The p ro jec tio n of a lin e segment PiPg on a lin e is

the segment determined by the p ro jec tio n of i t s end

points Pj and Pg on the l in e .

Thus, in the above figure is the p ro jec tion of

the lin e P%Pg upon the x—ax is .

The p ro jec tion of a lin e segment P^Pg upon a plane

is the segment determined by the p ro jec tio n s of i t s end

po in ts P^ and Pg upon the p lane. Thns, in the fig u re

MMg is the p ro jection of the l in e segment P^Pg upon the

xy—plane.

By p ro jec tion we mean here ^orthogonal pro je c tio i^ " .


5. The angle between two lines in space#— If the lines

in space intersec^ the angle between them is the angle be

tween their positive directions. I f the lines do not inter

sect, we may draw lines throngh the origin parallel to the

given lines and having the same direction# The angle be

tween these intersecting lines is the angle between the two

given non-intersecting lines.

6# F irst theorem of projection.— If ^ is the angle

between two directed lines and ^ in space, and

Pg are any two points o n /^ , then the projection of the

directed segment P^Pg on is

Proj/^PlPg « P^Pg cos G.

Proof : The projection of the point

p. on/-T is Pn and the projection of “T*. 5

the point Pg on is Mg. Hence the J

projection of P^Pg on Is P Mg# But

triangle P PgMg is a right triangle and

P H * P]Pg cos 0. But PiMg is the projection of P^Pg on

and hence

Proj£^PlPg « P^Pg cos 4.

7# Second theorem of projection#- The sum of the

projections on any l i n e i n space of the line segments


^1^2» ^2^3» ^3^4........ ^n—l^n» forming a broken lineis eq,ual to the projection of the line segment P^Pn onA

Proof; Let the coordinates of the points P^, Pg......be (3:1,71,Zi), , (xg^yg.zj),.... respectively,and let these points be projected upon the x—axis* Then

- Mil% - ig - i i

ProJxPg^g = 1%1% -

^^®^x?4^5 “ M4M5 = ZgT 4

I*roj]cPn-lPn “ “ = “^ -1

Adding these we have x^-^i which is the projection

of PiPji on the x—axis.

8. Point of division.— To find the coordinates of a point P which divides the line joining Pi(xi,yi,zi) and ^ 2 (^2 »2rg,Zg) so that P^P:PPg = m m , through Pi, P, Pg draw lines parallel to OX and meeting

tr». cthe yz—plane in Ui,U, and Hg respec— I /^* tively. In the plane determined by

I


and PgEg draw P Q and PR paralle l to lî îls* Then

%P^ - %i, HP » X, and HgPg = Zg, QP « and RPg =

xg—X* Triangle RPgP and triangle QPPL are similar since

respective sides are paralle l and therefore

OP H ± d l_PP2 RPg* ^

and X * mxg-t-nx]m+n *

Similarly y * Byg»ayi g = mzg+nzi ^m+n m+n

For the mid—point these formulas become, x «

y - 3[2|yi. z . f 2 ^ .

notice that we have these same formulas for the

coordinates of a point which divides a line in a given

ra tio in plane analytic geometry except there is no z.

Example » Find the coordinates of the mid—point of the line joining the points (3,4,5) and (—1,10,4).

Solution: Using the mid—point formulas

X = « 1, y = = 7, z » ^

and the coordinates of the mid—point are (1,7,^)

9* Distance between two points.— Let the points be

Pl(:Cl,yi,Z2) and Pg(xg,yg,Zg) and through P^ draw

P- M, and P Q paralle l to OZ, GY, and OZ respectively.

Through Pg pass planes perpendicular to these lines.


Then is equal to the projec

tio n of P^P2 on the x—ax is , P%M ^

is equal to the pro jection of P^Pg

on the y—ejcls, and P^Q is equal to

the p ro jection of Px^2 the z—a z is .

But P^R;* MS and hence

MS = Xg—Xx, PxM = yg^yx* PgS = Zg'-zx'

Also ÎE»x * + ^ x ^ t

* 5 is^ •

* (xg-x^)^ + (ygr-yx)^ + (zg-zx)^ ' (1 )

In p a r tic u la r , the distance from the o rig in to the

point P (x ,y ,z) is

OP * ■✓xS+yS+z . (2)

I t w ill be noticed th a t when zx “* Zg in (1) we obtain

the distance formula found in plane analy tic geometry; viz.

d^ • x^+y2.

Example 1. Find the distance from the point (11,11,-3) to the point (8 ,15 ,2 ).

Solution; Using formula (1) of the preceding paragraph

42 = (8-11)2 + (15-11)2 + (2+3)2 =. 50

and hence d * 5/?*


8

BXaniple 2* Express by an equation the fa c t th a t the point (x .y , z ) Is eq u id is tan t from the po in ts (1 ,1 ,1 ) , (—0 ,-5 ,7 I •

S o lu tion : The d istance from the po in t (x ,y ,z ) to thepoint (1 ,1 ,1 ) Is

i/ ( X—*1 ) 2+( ywX ) 2+( 25—1 ) 2

and the d istance from (x ,y ,z ) to (—3 ,—5,7) Is

V( x+3 ) +( y+ 5 ) +( z—7 )

and since these d istances are equal

■/(x—l ) 2+(ywi)2+( z—1)2 = y (x+3 )2+(y+5 )2+( z—7 )2

Squaring both members of th is equation we obtain

—8x-i-12y+12z « 80

as the requ ired equation.

Example 3. Find the equation of the locus of a point whose d istance from (2 ,2 ,-^ ) Is 11.

S o lu tion : Let the po in t be P (x ,y ,z ) . Then

11^ = (x—2)^ + (y—2)^ + (z+6)^

which Is the required equation.

10* Volume of a te trah e d ro n .— Let the coordinates

of the te trahedron be A(x3 ,y i,z ^ ) , B(xg,y2 ,zg ),

C(3C3,y3,Z3), L(x4 ,y4 ,Z4 ) and l e t

C ( 3 :3 ,^ ) . F"/'*?- 7(x4 ,j)4 ) be the p ro jec tio n s o f <J

the v e r tic e s upon the xy—plane.


9

]Totioe tha t we are supposlJag th a t D l ie s w ithin / / /

the triang le ABC.

Taking the co\mter—olookwiae d irec tion of the ver

tic e s of the trian g le s as positive we have

1 *1 71

Area A B D « 1?

1 ^2 ^2 (1)1

3 : 4 ^4

1 ^2 72Area » i

17 3 (2 )

1 ^4 7 4

17 3

Area JDC A - 2 1 ^1 y i (3 ) or

1 ^4 7 4

%1 3 1 y i

Area A B C * ? 1 % ^2 (4)1

7 3

1?

1 ^11

1= 4

yii I731 (2 )

Volume - ) ( 2 ) + ( ) ( 3 )

"^^1’*'*2‘*‘®3 d ) 3 " ^(z^^+zg+zg) (1 ) + (zg + zg + g ^ )(2 )

—(z£+Z3^Z4)(3'') — (z]^+zg+zg)(4)3 » ^ ( z^^+zg+Zg+z,^^) ( 1 )

+(z^+zg+zj+z^—z^5(2 )—(z^+zg+zg+z^—Zg)(3 )—(z^+zg+zg+z^—Z4 ) (4)3

Henoe we have for the volume

( z%+Zg+zg+Z4 ) ( 1 ) +( z^+zg+zg+z^ 2 )—( Z]^i^zg+Zg+Z4 ) ( 3 )


10

—( ( 4 ) — 2 )+zg(3 )—Zg(l)+z^(4) ] .

But the aum of the f i r s t four terms in the bracket

i s zero and henoe we have

1 X2 ^2 1

6T z^ 1 73 + Zg 1 yg1 3 4 74 1 *4 74

1

-►Z4 1 Zg ^21 73

and therefo re the volume is

— Zf

111

^1 y i

^4 ^4

1

1111

*1Z g

z

^ y i Xg yg

*4 74. ‘ 4

H . D irection cosines#— The d irec tio n cosines of a.

lin e are the cosines of th e angles th a t the upper end of

the lin e makes w ith the p ositive ends of the axes# I f

the angles made w ith the axes are denoted b y ^ ,

respectively* then cos , cos cos are the d ire c tio n

cosines of the line#

12. Relation between direction cosines,— I i e t b e

any line and. a line through the origin parallel to


I l

making the angles (Y, respectively with the coor

dinate axes* Project any point P(x,y,z) of upon the

x—axis forming the triangle OPQ,.

Then OQ, “ x, and OP • -/xZ+yS+zS *»r.

00.003 0( - OF - '

Similarly cos » Z, and

COS — #rBut COS Qj/ + cos^ (0 + cos^ ^ ^ ^

I1.

Hence cos^ c/ + cos^ (3 + cos^ ^ * 1.

13. The angle between two lin es .— le t ^ and be any

two lines through the origin whose direction cosines are

cosc^l, cos cos and cos ^ g , cos /^g, cos ^

the angle between the lines be G».

From a point Pj on &rop

a/ perpendicular P^Pg to Zg. le t ^ ^

0?2 * 2?x 01*2 = rg. Since OP

is the projection of OP on/^

Tg « r^ cos G * x^ooso^ y^cos ^+Zx®°s (i)

Dividing both members of the equation formed by the

second and th ird members of the equations (1) by r .


12

cos cosc^2 + cos 2 + ~ cos ^

But ^ « COSc^l, — a COS(^T, f i = COSr%

.Sind henoe

cos ^ ■ cos cY iCOS Y s + 008 I^COS ^2 + 008 j^^COS ^ (1)

I t follows a t once from eq.uation (1) th a t the condition

fo r the lines to be a t rig h t angles to each other is

cos <?/ cos«7{ 2 + cos ^^00 8^2 ■*’ ®osJ^2.®°®^2 “

and the condition for p a ra lle l i ty is c learly

9^1 - °<2 . 4 . ='^2* “ ^2-14# Direction cosines of a line through two points#—

le t the points be and

and the distance between * ^ 3 ' ^

the points r . Denoting the d irection

cosines of the line joining and ?£

by cos <t{, c o s cos ^

ctosT^ a —Z—, oos lf =yi-yg «1-Z2

Henoe the d irection cosines are in the ra tio s

/Z e ro is e s

1# Find the length of the segments having the end points (7 ,3 ,5 ), (4 ,-9 ,21; (-1 ,3 ,2 ) , (5 ,3 ,-2 ); (3 ,-7 ,9 ), (6,9 ,-1)#

2# Find the coordinates of the mid—points of the segments in Exercise 1*

3# Find the coordinates of the point dividing the segment


13

A (3,2,5), B{4,—5,7) in the ra tio 1:2*

4. Find the d irec tion angles of the line which makes equal angles with the coordinate axes*

5. What is the distance from the point {—3,4,5} to the point (3,—4 ,—2)?

6* Find the coordinates of the poihts a t a distance 5 from the points (—2,—2 ,1 ), (3,—2,6) and (3 ,3 ,1 ).

7. Find the d irec tion cosines o f a lin e which makes an angle of 60° with the x—axis and equal angles w ith the y and z axes*

8* Find the d irec tion cosines of a lin e which are proport io n a l t o '3 ,—4, and 2*

9* A lin e makes an angle of 60® with the x—axis and 45® with the y—axis* What angle does i t make with the z-axis?

10* I f the projections of FiPo on the axes are respective ly 3 ,-2 , and 7 and i f the coordinates of sne (—4 ,3 ,2 ) , find the coordinates of Pg*

11* A broken line jo ins continuously the poin ts (6 ,8 ,—3 ), (0 ,0 ,—3), (0 ,0 ,6 ), (—8 ,0 ,2 ), and (—8 ,4 ,0 ). Find the sum of the pro jections of the segments and the p ro jection o f the closing line on (a) the x -ax is, (b ) the y—ax is , (c) the z— ax is .

12. Sho^ th a t the points (—3 ,2 ,—7), (2 ,2 ,—3), and (—3 ,6 ,—2) are the vertices of an isosceles tr ia n g le .

Group B

1. What is the value of one of the d irection cosines of a lin e p a ra lle l to the xy—plane? the yz—plane? the zx—plane? What re la tio n ex is ts between the other two?

2. Find the distance from the point (1,2,3) to the point (2 ,3 ,4 ).


14

3. Find the lengths of the medians of the triangle whose vertices are the points (3,4,—2), (7,0,8), and (—5,4,6).

4# Find the eqaation of the locus of a point which is equally distant from the points (3,2,—1) and (4,—3,0).

5. Find the equation of the locus of a point which is three times as far from the point (2,6,8) as from the point (4 ,-2 ,4).

6. The direction cosines of a straight line are proportional to 1, 2, 3; find th e ir values.

7. A straight line makes an angle of 30® with the x-axis and 75 with the z—axis; what angle does i t make with the y—axis?

8. Show why i t is that a line cannot make an angle of 30® with the x-axis and an angle of 450 with the y—axis.

9. Show that the point (—1,-2 ,—1) is on the line joining the points (4,—7,3) and (—6,3,—5) and is equally distant from them.

10. A broken line joins continuously the points (6,0,0), (0,4,3), (—4,0,0), and (0,0,8). Find the sum of the projections of the segments and the projections of the closing line on (a) the x—axis, (b) the y—axis, (c) the z-axis.

11. Show that the points (4,3,—4), (—2,9,—4), and (—2,3,2) are the vertices of an equilateral triang le .

12. Prove that in any quadrilateral the sum of the squares of the diagonals is double the sum of the squares of the lines joining the middle points of the opposite sides.

Group G

1. What are the direction cosines of a line paralle l to the x—axis? to the y—axis? to the z—axis?

2. Find the distance from the point (3,—8,6) to the point (6,—4,6)•

3. Find the coordinates of the points dividing the line from (—2f«p.) to (3,-2,4) externally and internally in the ra tio 2:5.


15

4* Find the loons of a point which is eqnally d is tan t from the points (S ,4,—1), (3,—5,0).

5* Find the eanations of the locus of a point which is equally d is tan t from the points (6 ,4 ,3) and (6,4,9) and a lso from (—5,8,3) and (“ 5,0,3/#

6# Find the d irec tion cosines of a lin e i f they are proportio n a l to 3, ,6 , and 2.

3 5 2.7. Show tha t the lin es whose d irec tion cosines are y, y# y#

7 ' — y; — y» y» y are mutually perpendicular.

8. What is the length of a lin e whose pro jections on the coordinate axes are 4, 1, 3 respectively?

9. Find the pro jections on the axes of the lin e joining the o rig in to each of the points (—3 ,4 ,—8), (5 ,—6 ,4 ) , (8 ,6 ,0 ).

10. Prove an a ly tica lly th a t the s tra ig h t lin e s joining the mid—points of the opposite sides of any q u ad rila te ra l pass through a common point and are b isected by i t .

11. Show tha t the points (3 ,7 ,2 ), (4 ,3 ,1 ) , (1 ,6 ,3 ) and (2 ,2 ,2) are the v e rtices of a parallelogram .

12* Find the coordinates of the point of in te rse c tio n of the medians of the tr ia n g le whose v e rtices are the points Pl(x3^»yi»zil), I*2(3C2,y2,Z2) and Fg(3:3,y3,Z3).

Miscellaneous Problems I - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1. Prove th a t the lin es joining the mid—poin ts o f the opposi te sides of any q u ad rila te ra l in space b isec t each o ther.

2. Find the locus of a point P (x ,y ,z) which moves so th a t i t s distance from the point F2(% i,y i,z i) i s always constant.

3. Prove tha t the lin es joining the v e rtices of a te t r a hedron to the points of in te rsec tio n of the medians o f the opposite faces meet in a point which divides th is lin e in the ra tio 3:1.


16

4* Prove that the lines which join the mid—points of the opposite edges of a tetrahedron meet in a point which bisects each line.

5. The line joining to (zg.yg.zg) is dividedexbernally in the ra tio nw-n. Find the coordinates of the point of division.

6. Find the area of the triangle whose vertices arePsUs.yg.zg).

7. I f P is a fixed point on a straight line throngh the origin equally inclined to the axes, any plane throughP w ill intercept lengths on the axes the sum of whose reciprocals is constant.

8. Find the shortest distances between each of the diagonals of a parallelopiped whose edges are a, b, c and the edges which i t does not meet.

9. Prove that the points (a ,o ,o), a ,0 ,0), (0,a^?,0),( are the vertices of a regular tetrahedronand find the distance between the middle points of two opposite edges.

10. Prove that the shortest distances between the diagonal of a rectangular parallelopiped and the edges which i t does not meet are —iË, iË where the lengths

6of the edges are 3,4,5.


17

Chapter II

The Plane

15. Every equation of the f i r s t degree represents

a plane.— A plane is a surface such that i t contains

every point on a straight line joining any two of i ts

points; such that a ll of i ts points are not in a straight

line; and such that i t does not contain a l l points of

space.

Let and be any two points

on the surface whose eqmtion is

A z + B y + C z + L - 0 (1)

Then Az By^ Cz^t L = 0 (2)

and Azg+ Byg+ Gzg+ D = 0 (3).

If Pg(z^,y3,Z3) is any point on the line joining

and Pg and cutting P]Pg in the ratio nn-n, then

2 » S l H l . v_ = W Z *^ 1 . mz2+nzim+n o m+n * m+n

Substituting these coordinates of the point Pg for

X, y, z in the le f t member of (1)

* b(12 ) . c(2m )

m""+"n( - a- Byg+Czg+L) + —^ ( AZg +By +Cz +B )

w hich i s equal to zero s in c e by (2 ) Az^+By^+Oz^+D * 0

and by (3 ) Azg+Byg+Czg+D =*0.


18

Therefore Pg, any point on the line P^Pg is on

(1 ) .

Values of x, y, z may be found, such that equation

(1) is sa tisfied for any given set of values of A, B, C,

D and such that the point x, y, z does not lie on this

line and hence (1) is not a straight line.

Moreover, for ary fixed set of values of A, B, C,

P values of x, y, z can be found such that equation (1)

is not sa tisfied and hence {1} does not contain a l l points

of space.

Since the three conditions imposed in the definition

of a plane are sa tisfied

A x + B y + C z + D * 0 is the equation of a plane.

16. Equation of a plane through three given points.—

le t the three points be *

The general equatiop^of the plane is

A x + B y + C z + D “ 0 (1)I f the three points are on the plane

Byi+ CZ2+ P * 0 (2 )

AXg+ BV2+ Czg* P » 0 (3)Azg+ Byg+ Czg+ P » 0 (4)


19

(5)

Eliminating A, B, c, D from equations (1), (2),

(3), (4) we have

X y z 1

7 1 z i 1- * 0

Z2 72 Z g 1

^3 73 Zg 1

which is the equation of a plané since i t is of the f i r s t

degree* I t passes through the three given points since

substitu ting the coordinates of any one of the points in

the le f t member of (5) makes two rows identical, and the

determinant is therefore equal to zero. Hence (5) is the

equation of a plane through the three given points.

Example. Find the equation of the plane passing through the points (3 ,1 ,1), (-2 ,^ ,3 ), (-3 ,4 ,1 ).

Solution: Substituting in the determinant above for X q ,2% e tc ..

X y z

3 1 1

-2 I 3

1111—3 4 1

from which we have

3x + 6y + 9z + 2(J * 0

which is the required equation.

17. Intercept form of equation of plane.— Let the

A x + B y + C z + D » 0


20

meet the axes in the points ( a ,0,0 ), (0 ,^ ,0 ), (0, 0 , c ).

Then by 16 the equation of the plane through these

three points is

X

a

00

y

0

b

0

z

0

0

c

1

1

11

from which we obtain

box + acy + abz — abc = 0

Which is the in tercep t form of the equation of the plane.

Compare th is with the in tercep t form of the eqaation

of the s triag h t lin e in plane analy tic geometry, viz

I * Ç -To obtain the in tercep t form of the eqaation of a

plane divide both members of the eqaation by the constant

term-B, and make the coeffic ien ts of x, y , z un ity .

Example. From the equation 6x—5y—4z+3 = 0 obtain the inte rcep t form of the equation of th is plane.

Solution: Rewriting the equation we have

6x —• 5y ~ 4z “ —3.

Dividing both members of the eqaation by —3


21

—2% + + ^ •* 1

or » 1.^ 3 ?

Example 2* Find the equation of the plane whose in tercepts are —1, —1, and 4,

Solution; The intercept form of the equation of a plane ia

2 + Z + z - 1 .a b c

In the given example a«—1, b*—1, c«4*

Hence we have

=T- * 4 “ * I ■ ^and. 4x + 4y — z •» —4.

18. normal form of equation of plane.— Let p be the

length of on the perpendicular

from the origin on the plane,

and le t the direction cosines ?

of on be cos c , G0 8 ^ t 008

le t P be any point on the plane

with coordinates x = OM, y ? LM,

z ** IP. The projection of the broken line omp on on is

equal to the projection of OP on on. The projection of

OP on on is on. The projection of OM on on is x 005 c( ,

that of LM on On is y cos /3, and that of IP is z cos .


22

Hence x cos^ + y cos z cos ^ p (1)

which is the normal form of the eqLnation of the plane.

The general equation of the plane

Ax + By + Cz + D * 0

can be put in the normal form.

In the normal form of the equation of a plane the

coefficients of the variables are the direction cosines

and since any three real numbers are the direction co

sines of the same line i f the sum of the ir squares is

unity, i t follows that

Ax By Cz _ —D /oiv x K s K c i vaSVbH^^S yAS*B2+ca

is the normal foim of the equation of the plane since

( ^ ( — ■ ■ _______ + {■ P a 1 ,✓a2+B +c2 yA2+B2+c2 VA2+B2+c2

Example. From the equation 4x—5y^z = 9 obtain the normal form of the equation of th is plane.

Solution; In th is example A » 4, B “ —5, C = —6, andhence -/A +B Vc “ ‘/4^+ ( —5 ) 2+( i— 6 ) 2 * ✓VV.

Hence substituting in (2) above

Ax____5y 6z , 97 W 7 W 7 W

which is the normal form of the given equation.


23

19. Equation of plane through the intersection of

two given planes.— Let the equations of the given planes

beAoc + By + Cz + D= 0 (1)

0^2+ Dq = 0 (2 )

Then the equation of the plane through the in te r

section line of planes (1) and (2) may be written in the

form

Ax+By+Cz+L+kCA^x+Bj^y+C^z+Lj^) « 0 (3)Since any set of values satisfying both (1) and (2)

w ill satisfy (3) and since (3) is an equation of the

f i r s t degree in x, y, z which may be written in the form

(A+kA lx + (B+kB2)y + (C+kC] )z + (D+D ) ■ 0

where A, B, C, D, A e tc ., are constants, (3) is the

equation of a plane.

Example. Find the equation of the plane through the inr- tersection of the planes 2x+y-z « 4 and x-y+2z = 0 and passing through the point (0,—1,1).

Solution: The equation of a plane through the intersection of 2x+y—z = 4 and x-y+2z » 0 is

Ex+y—z—4+k(x—y+2z) « 0 (1)

Since the plane is to pass through the point (0,—1,1) th is point must satisfy equation (1) and hence

—1—1—4+k(l+2) » 0 and k * 2 .


84

Substituting k = 8 in (1)

4z-y+3z—4 * 0

whioh is the required equation.

80. Parallel planes.— Two planes whose equations

are

Ax + By + Cz + I) ■ 0

and Ax + By + Cz + 3) « 0

are paralle l for a l l values of sinoe A, B, and C are

proportional to the direction cosines.

Example. Find the equation of a plane which passes through the point (3,8,—1) and is para lle l to the plane 7x—y+z*14.

Solution; The equation of a plane paralle l to 7x—y+z * 14 is

7x-y+z * k (1)

I f the plane is to pass through the point (3,8,—1)then th is point must sa tisfy equation (1) and hence

81 —» 8 — 1 * k k » 18.

Hence the equation of the required plane is 7x—y+z » 18.

81. Condition that three planes have a common line

of in tersection .- Let the equations of the three planes

beA x + B y + C z + B = 0 (1)

A%x+ Biy+ C^z* * 0 (8)

Agx+ Bgy+ Cgz+ Bg * 0 (3)


25

I f the three planes pass through a line then (3)

w ill pass through the in tersection of (1) and (2) and

may he w ritten

Ax+By+C z +D+k ( A x+B y+C z +D21 =* 0 ( 4 )

and therefore - 2î ^ . _ ^As “B i^ —C|“ ~IÇ ~

Putting each of these fractions equal to —r we

have

A+kA +rAg • 0

B+kB^+rBg » 0

C+kCi+rOg « 0

m-kP-j +rDg = 0

Taking these equations in groups of three and elim

inating k and r we have

A Al 2 A ^2 B ^1 A Agi

B % 2a 0 C Cl = 0 G Cl ( 2«0 B ®1 »2C Gl B 2 D ? 1 % ^1 ®2

or changing rows into columns these conditions may he

w ritten in the form of a matrix

A B C B

^1 % Cl

^2 Cg ®2 1

which is the condition th a t the three planes sha ll pass

through a common line#


26

22. Angle between two planes.— The angle between

two planes is the angle between two lines perpendicmlar

to them.

The direction cosines of the perpendiculars on the

planes

Ax + B y + C z + D « 0

and A x+B y + C z+ = 0

are respectively

A B Ci/A +B +C VA +B +C yA +B +C

° i .VA 2+B 2+c^S •i A +B] 2+q 2

Denoting the angle between these perpendiculars

by ©•, we have / 13AAm BBn

c o s ^ *• — — — —-— ---------— -■ — ■ - — -yA2+B2+c2VAi2+Bi2^Ci2 yP+iS+^VAx^+B^S+CiS

. CC, AAi + BBi + CC,VA^+B^+C^Ax2+Bx2+Cx2 yA^+B^+C^^A^S+B^^+c^^

I f the planes are at right angles cos ^ *= 0 and

AA BB* + CCx ™ 0.

I f the planes are paralle l


27

i l “ I l ■ 11"

23. Distance from a point to a plane.— Let

be the given point and

X cos + y cos 2 cos ^ “ p (1)

be the equation of the given plane.

Then x coso + y oos(S + z cos (2)

is the equation of a plane parallel to (1) and i f (2)

passes*through the point PiCx^jy^^^z^) then

x^cosof + y^cos^ + z^cos ^ = p-j

and the distance from the point p is the distance between

the two paralle l planes (1 ) and (2) and hence the distance

from the point p to the given plane is

p^—p = X] 0 0 8 of + y^cos z^cos ^ — p.

If the equation of the plane is in the form

A x + B y + C z + D = 0

we obtain the normal form of th is equation, transpose

a l l terms to the le f t member of the equation, and then

substitute the coordinates of the given point (x2,y%,Z2)

givingAxi + By + Cz%+D

VA2+B2+C2


28

which is the distance from the point to the

plane Az + B y + C z + D » 0 *

Exercises

1. Find the intercept form of the following equations:(a) Ex+3y+5z » 30 (h) 2y—x--®z ** 5(c) 5x+4y—2z+3 * 0(d) X—y+z = S

2# Find the equations of the planes having the following intercepts

(a) 1 , 2

(h) —3, 3, —1

(o) i , - i . 2 U ) - | . - I

3. Transform the following equations to the normal form.(a) 3x+2y+7z = 20(b) X—3y+z = 10(c) 4x—3y+2z+15 = 0(d) 7x—5y+3z+l = 0

4. Find the distance from(a) x+y+z ** 10 to the point (2,3,5)(b) 3x+4y+z * 4 to the point (5,3,1)(c) 2x+3y—6z s* 7 to the point (-1,3.-^)(d) X—2y+3z » 5 to the point (2 ,2 ,l |

5* Find the equations of the planes through the following sets of points.

(a) (1,0,0), (0,2,0), (0,0,3).(b) (1,2,3), (2,3,1 , (3,1 ,2).(o) (1,1,1), (2,3,3), (3 ,1 ,1).

ReprocJucecJ with perm ission of the copyright owner. Further reproctuction prohibitect without perm ission.

29

6* Find the equation of the plane through the originp a ra lle l to the plane 3x+4y+6z+5 » 0.

7* Find the e<iuation of a plane p ara lle l to the plane * 14 and two un its farther from the origin.

8. Find the distance between the planes 2x—2y+zs—18 « 0and Ex—Sy+z—12 « 0.

9* Write the eq.uation of a l l planes p a ra lle l to the plane 7x—3y+5z » 14.

10. Determine k so that the plane x+Sy—kz = 9 shall pass through the point (3 ,6 ,3).

11. Find the coordinates of the point of in tersection of the planes x—2y=8, x+2y+z “ 0, and x+y+z = 3.

12. Show that the plane x+Ey—2z « 9 passes through thepoint of in tersection of the planes x+y+z= 1, x-y—z * 1, and 2x+3y « 8.

24. Eq.uation^of a stra igh t l in e .— Since the coor

dinates of any point on the in tersection line of two

planes sa tis fy the equation of each of the planes, two

in tersecting planes determine a stra igh t line and i t

follows then that two. equations of the f i r s t degree

represent a stra igh t line*

25. Direction cosines of in tersection line of two

planes.— Let n be the direction cosines of the

line of in tersection p^ of the two planes

Ax + By + Cz + D, » 0 (1)

AgX+ Bgy+ Cgz+ Dg = 0 (2)


30

Xat p be a line perpendicular to (1 ) and pg a

line perpendlanotlar to (2). The direction cosines of p

are proportional to A, G and the direction cosines

of pg are prapertlonal to Ag, Bg, Cg. But p and pg are

both perpeaftiealar to the line p and henoe

Âê * ®l“ ® (@)

yÂge mBg+ nCg" 0 (4)

SolTing aquations (3) and (4) simultaneously for

ân d # n n

^m :n •

by R

we

Bi Cl Ox Ai BxBg Cg * 0 2 Ag * Ag Bg

Denoting

have /

4p i 0^6 |0l A3]g ^1 ®ljBg Cgj Og Ag ^ ®gBx Ox Cl Ax Ax ®X2 s_2s m ml £ s ^ n «»

26* Condition that three points are oolllnear*— If the

three points Pi(xi»yi#*i). » 1 3 7 3 , 2 3 ) are

oolllnear, the direction cosines of the line P^Pg will be

the same as the direction cosines of the line PgPs* That

is

^-3[g2yi-yg:zi-2:g - ^cg-xgiyg-ygrzg-zg (1)^1 . yi-yg _ »1 " ^2 /p%Zg - *3 yg-y3 - 23

or


31

"but th@ denomimtors are proportional to the direction

coaines Ana hence ( 2 ) may be written

^1 - ^2 . yX-" ^cos ^ C O S ^ C O S ^

whioh is the condition that three points are collinear#

27. Equation of a line through two points.— Let

the two points be and and let

P be any point on the line joining P% and Pg. Then from

26 i t follows that

xg yx— yg zx— zg

and since the denominators are proportional to the direc

tion cosines,

i , - . a . I— I} . . r . - , =1 (1 )cos COS 008 f ' ' 'I f the direction cosines are proportional to jf, m, an

n, then ( 1 ) becomes

^ " ^ ^ 1 . y ^ y i a, ^ - ^1

Example. Find the equations of the line joining the point (-3,2,1) and (6 ,-4 ,-2 ).

Solution: Using the results just obtained

or - 2 -


3S

ZBm Farp endioolar dlstanea of a given point from a

given straight llna*— Let Fg be the given point and F%

a point on the given line* The agnation of the line le

i j u a - y - y i _B n

Throngh and Pg pass planes parallel to the ooor

dlnate axes so that P B, R8 , and

SPg are edges of a parallelopiped.

The projections of P Pg on the J I ^line P Q and is egnal to the earn of

the projections of the broken line

P1 BSP2 ( see ^ 7 ).

p^R = ♦ (ysrSTi)™ ♦

- 1 ^ .

- (Xg-z^)2+(y2^^)2+(z!2-z^)2

and hence

— [(jCg-Xi)/+(y2-yi)»*{z2*^l)«»]2

which is the square of the distance from the point

zg #72 to the line through PP *


33

29. Condition that two straight lines intersect.—

Let each of the two lines be represented by two planes.

Then the condition that the two lines intersect is that

the four planes pass through a common point* Let the

lines be represented by

* 0 )

AgX + Bgy ♦ CgZ + Lg *and

A3X + Bgy + Cgz + Dg

A4X ♦ B^y + C4» ♦ T>

The condition that these four planes pass through

a common point is found by eliminating x, y, and z from

the four equations aboTe. This condition is expressed

by the equation

Cl Bia.Ai «1

Ag #2

A3 B5

0 .

^4 -°4 '^4 "4

If the equations of the lines are in the form

y-y:m

and the lines meet, they lie in the same plane and

since the plane passes through the point (xi,yi,zg^)

we may write its equation


34

A ( + C( z—z^) « 0 ( 1)

But the point la also on the plane

and henoe

A(Zgr^i) + B(yg^2) ♦ C(zg-®3 ) * 0 (2)

Sinoe the normal to the plane is perpendicular

to each of the lines we have

A/^+ Bm + Cn * 0 (3)

A/^+ Bm * Cn% * 0 (4)

Eliminating A» B, C from equations (2), (3), (4)

we have

m

®1

zgrzin

“1

which is the required condition*

30* Plane through two in tersec ting lines*— In

29 we found th a t I f two lin es

X—ac~ j ~ ■

in tersect» then

MTim

xgf-xi yg-yq Z2~*l

^ m nm<i=1 “l

and therefore the plane contains both lines and hence


35

jLy ^ im

/■1 “ x

is the plane determlned hy the two l in e s providing they

in te rsec t*

31. Plane through lin e p a ra l le l to given l in e .—

Let

Z 2 1 z—s- “n" ( 1 ) and

Z-gg

“1(21

be the two lines.

Then the line y-3Timi

Z— Z ’ ^1

(3)

r-OCl z—zd m n

A “ 1 ^ 1

meets (1) in the point (x i#y i»z i) and is p a ra l le l to (2)

sinoe i t has the same d ire c tio n cosines.

Prom ^30

“ — “ 1- 0 (4)

is the equation o f th e plane through the in te rse c tio n

o f (1) and (3 )t and th e re fo re (4) is the equation o f

the plane through (1) which i s p a ra l le l to (2 ) .

32. Plane throngh a po in t p a ra l le l to two given

l in e s .— Let (zg.yg^z^) be the given poin t and


36

(1) and“1

Z—12^ (2 )

the given lines*

Then x-»3"Z "

y^5

and

a-*5n

z-«3»1

(3)

(4)

are lines through (xg^yg.zg) and parallel to (1) and <2)

respeotirely*

Henoe

IMT3m

a-zg

n - 0 (5)

“1 “1

is the equation of the plane through (3) and (4) * It

is parallel to (1) sinoe (5) passes through (3) and (3)

is parallel to (1) and (5) is also parallel to (4) since

(4) is parallel to (2)*

33. Line parallel to plane*— If a line is parallel

to a plane then the normal to the plane is perpend!ouhar

to the line. Let the equation of the line be

x-xi y-sTi Z—

and of the plane

Ax + By + Cz + ] )« 0 .


37

Then ùk. ♦ mB ♦ nC » 0

is the Qondition th a t they are p a ra lle l*

I f the equations of th e l in e i s represen ted by the

equations

+ B] y ♦ " 0

Agx + Bgy + Cgz + Dg - 0

then the condition fo r p a r a l le l i ty Is th a t the th ree

equations have no common solution* That I s ,

A B C

Ai Bx Cx

34* Plane through l in e perpendicular to given plane

Let the equation of the lin e be

x-oci _ y-y^ _ z-cx m “

and of the plane

Then

( 1)

( 2 )

(3)

Is the equation of the lin e through Zx,yx»^l Perpendicu la r to the given plane*

But X—Xt y ^ i z—Zxm n « 0 (4)

A B C


38

la the equation of the plane through (1) and (3), anA (4)

eontalns (3) vMeh la perpendloular to (2) and hence (4)

la perpendloular to (2)«

Equation (4) la the equation of a plane through (1)

perpendicular to (2) and henoe equation# (S) and (4)

represent the line which Is the projection of (1) on (2)*

35* Shortest distance hetween two lines*— Let the

equations of the lines be

(1) ^

By 5 32

y^g“1

s—z.(2)

?/ l

r-y im“1

n

“l(3)

is the equation of the plane containing (1) and parallel

to (2) and

y^2 z-Zg ^ m n « 0 (4)£ l n^

la the equation of the plane containing (2) and parallel

to (1). The shortest distance between the two lines Is

obviously the distance between the parallel planes*


m

Obtaining the normal form of (3) and. (4) and talcing

the d ifference of the d istances and pg of th ese

planes from th e o rig in we s h a ll have the requ ired

d is tan ce•

W riting the equations in the normal form

“1mi n i

®1 »1

7

^1

y i 5=1 m nm-i n-i

and

m n |2 n ami n^ + ^ 1 ^ 1 +

y iz

" Z i mi>m n

/ mi n i

Sim ilarly p

yÇ- “ r + U “ I' L^i “ a / i “ni

*3 7 zm

/ l mi— —

/ m i n j * | / i n j“1 “1

and the required d istance is

P i - Pg

*1 y i *1yC m n1 Hi* \ a ____________

7m n )3y mi n j +

I

yg

z A

7/ l m i _________

“ 4 :J

*2aHI


40

/ | " i “J * A

Sxeroiaes.

1. Find the direction eoslnes of the following lines.(a) Zjc ^ Z z • 0

x+2y^z ■ 4

(h) • 5x-y® - 3

(e) x+y * 6 2xH3y * 5

2. Show that the following pairs of lines are paralle l.(a) 2y+8 « 0, 3y-4a *» 7, and 5y-2a * 8, 4y+llz * 44.(h) 3x+z « 4, y+2z ■ 9, and 6x-y ■ 7, 3y+6z ■ 1.

3. Show that the following pairs of lines are perpen— dieular.

(a) x+2y * 1, 2y-z • 1 and x-y « 1, x—2z » 3.(b) 3x+y-z * 1, 2x—a * 2, and 2x-y+2z • 4,

x-y+2z « 3.

4* Find the equations of the llhe passing through the points (-2,2,1) and (—8,5,—2).

5. Find the equations of the projecting planes of the following lines.

(a) 2x+y« » 1, X—y+z • 2(h) 2x-y+z • 0, 4x+3y+2z = 6.

6. Do the following sets of points lie on a straight line?(a) (3 ,2 ,-4), (5,4 ,-6), and (9,8,-10).(b) ♦3,0,1), (0,-3,2) ana (6,3,0).(o) (2,5,7), (-3,8,1) ana (0,0,3).


41

7. yind the equations of the l in e passing through the point (2 ,—1,-3 ) whose d ire c tio n cosines are p roportional to 3 ,2 , and 7*

8. Find the equat ions of the lin e passing through the point (0 ,-3 ,2 ) which i s p a ra lle l to the lin e jo in ing the poin ts (3 ,4 ,7 ) and (2 ,7 ,5 ) .

9. fin d the angle hetween the lin e s»nit 2EJË * Ç i f both lin e s are d irec ted upward.

10* Find the aquations of the lin e passing through the point (3 ,—1,2) which i s perpendicular to each o f the l in e s X * 2s—1, y • s+3, and ^ ^

11. Show th a t the lin e ^ i s p a ra l le l to the plane 4x+2y+2* • 9.

12. Find the equation o f the l in e passing through the point (3 ,2 ,—6) which i s perpendicular to the plane 4*-y^3a * 5.

Exercises Group A

1. Find the angle between the two lin e s whose d irec tio n

cosines are *“ "y* ^ cud — re sp ec tiv e ly .

2. Show th a t the po in ts (7 ,2 ,4 ) , (4 ,—4 ,2 ), (9 ,—1,10) and (6 ,—7,8) are the v e rtic e s of a square.

3. Find the equation Of the plane which is perpendicular to the lin e jo in ing (3 ,4 ,—1) to (6 ,2 ,7 ) a t i t s middle p o in t•

4* Write the equation of the lin e through the poin t(1 ,2 ,1 ) perpendicular to the plane 3x—2y+6z—7 « 0.

5. Find the angle between the planes 2x+y—2z—9 « 0 and X—2y+2z » 0.

Find the q u a tio n of the plane which passes through the points (2 ,3 ,0 ), ( -2 ,-3 ,4 ) and (0 ,6 ,0 ) .


42

«qtiatlon of the plane whose Interoepts of the axes are 3, 5, 4«

8* Find the eqiiatlonf Of the line through the point (1,—2»3) parallel te the line

7x-4y^m+16 # 0 x+y-4mel8 ■ 0

9* Find the dlatanee het#een the parallel planes 6ie;^«-3»p-63 « 0 and 6x+8y^s+49 • 0*

10* Find the equation of the plane whioh Is perpendicu lar to the line

?x##y^$el4 " 0 xey^zeld # 0

and whioh passes through the point (1,—1,4).

11* Show that the plane x+gy-Sw-4 - 0 passes through the point of Interseotlon of the planes x*y+»-l * 0 X—y'-®-l * 0 , and 2x+%r-8 ■ 0.

12* Show that the four planes x+2ye2n^ * 0, x*F-2z+2 ■ 0, x-y+8 * 0, and 3x-y-2zel8 * 0 pass through the same point.

13. Find the equation of the plane which passes through the point (0,0,4) and is perpendicular to each of the planes 2x-3y * 5 and x-4z « 3*

14. Show that the four points (2,—3,4), (1,0,2), (2,—1,2) and (1,—1,3) lie in a plane.

15. Find the equations of the faoes of the tetrahedron whose vertioes are the points (0,3,1), (2,-7,1), (0,5,—4), and (2,0,1).

16. Find the equation of the plane whioh passes through P l(x i,y i,2i) and Is parallel to the plane ix ♦ By + Cz+D"0.

17. Find the equation of the plane which passes through the origin and P%(xi ,yi,z%) and is perpendicular to the plane Ax+By+Oz+D * 0.


43

18* Find the distance from the plane 4x+3y+lBs-f6 » 0 to the point

19* Find the locus of points which are equally d istan t from the planes ■ 0 and 8%-Sy+2z+4 « 0*

20* Find the TOlume of the tetrahedron whose vertioes are (3 ,4 ,0 ), (4,—1 ,0 ), (1 ,2 ,0 ), and (0 ,—1,4)*

21. Find the equation of the plane which makes an angle of with the plane y+s > 7 and passes through the points (0,—1 , 0 ) and (0 , 0 ,—1 )*

22. Find the leeus e f » point which i s 3 tim es as fa r from the plane 3m-6y-as * 7 as from the plane 2n-y+2z»9.

23. Find the equation o f th s plane whioh passes through the In terseotlon of the planes 2x+y-4 - O and y+2% » 0 andIs perpendicular to the plane 3%+2y-3z - 6 .

24. Find the equations of the planes passing through theIn tersection of the planes Aix+B y+Cj z+D] * 0 andAg%+Bgy+0gz+3g ■ 0 which are perpendicular to the coordinate planes.

Group B

1. Find the angle between the two lin e s whose direction oosinea are 3 _ & and 3 ,_ 2 , » resp ee tlv e ly .

2. Show that the points (7 ,3 ,4 ), (1 ,0 ,6 ), and (4,5,-2) are the vertices of a righ t tr ia n g le .

3. Show that the four points (2 ,-3 ,4 ) , (1 ,0 ,2 ), (2,-1,2) and (1 , - 1 ,3) l ie In a plane.

4* Find the equation of the plane which passes through the point (3 ,—1, 6 ) and is p a ra lle l to the plane X—2y—3z>4 « 0 .

5. Write the equation of a l l planes p a ra lle l to the plane 7x-8y+3z+2 * 0.


44

6* What is the angle between the planes 3x+5y—6z+l = 0 and 3x—Sy~6z+1 * 0?

7. Find the distance from the point (1,5,9) to the plane 3xr-5y—6z+l » 0.

8* Find the equation of the plane throngh the linex+y+z - 0 2x+y+z* 1

which passes through the point (1,—1,1).

9* Is the point (6,2,1) on the straight line determined by (2,4,3) and (~1,6,-1)?

10« Flnd^the equations of a line passing through the points (6 ,2 ,—1) and (3,4,—4).

11. Find the direction cosines of the line of intersection of the planes 3x+y—7z * 6 and 2x+3y+4z * 7.

12. Find the angle which the line3x+2y+7 = 0 2y+4z+3 * 0

makes with the plane x+y+3z-^ * 0.

13. Find the locus of points equally distant from the planes x+2y+3z+4 « 0 and x—2y+3z—5 * 0 .

14. Find the equation of a plane through the points (1,—3,1), and (3,—1,3) which is perpendicular to the plane &g^y+6z—14 * 0.

15. What are the equations of a line through (1,3,—5) p ara lle l to the line

y = 3x—14 7x—2z =17?

16. Find the point of Intersection of the lines3 x -y ^ z ^ * 0 x+y-z « 0x-y+z+2 • 0 6x—6y—3z—15 = 0

17. Prove that the three planes x+2y—3z+l = 0 , y+5z—1=0, and 3x+4y—19z+5 = 0 pass through the same s t r a i ^ t line .


45

18. A lin e Is drawn through the p o ^ t (1 ,2 ,3 ) having d irec tion angles c/ • 45 , ^ * 60®, » 60". Find thecoordinates of a point upon th is lin e 5 u n its d is ta n t from the poin t (1 ,2 ,3 ) .

19. Show th a t i f the lin e s

7 - ax+b and ^ “ ^1^+^^z * oz+d z = c^z+d^

in te rse c t, thenb— ^ d—dx a—&X c—@2

20. A point moves so th a t the snm of i t s d is tances from any number of given planes i s constan t. Find the locus of the po in t.

21. ^A(^2 »y2 »zg) a re any two po in ts.Find the locus of a point Q in the ^qr^lane so th a t the angle PxQPg 1» a rig h t angle.

22. Show a n a ly tic a lly th a t the locus of po in ts equally d is tan t from any th ree given po in ts i s a s tra ig h t line perpendicular to the plane of the th ree p o in ts .

23. I f the normal d istance from the o rig in of the plane which makes the in te rcep ts a, b , c re sp ec tiv e ly on the z , y , and z—axes i s p, prove th a t

1 « 1 ^ 1 . 1 p2 â2 b2 Ô8*

24. Find the center of a sphere of rad ius 7 passing through (—3 ,1 ,—9) and (1 ,—5,3) and tangent to the planes 2zv3y+6zi-23 « 0.

group C

1. Find the angle between two lin e s whose d irec tio n cosines are | and | re sp ec tiv e ly .

2. Show th a t the po in ts ( -6 ,3 ,2 ) , (3 ,—2 ,4 ) , (5 ,7 ,3 ) and (—13,17,—1) are the v e r tic e s of a trapezo id .


46

Show that the four points (1,0,—1), (3,4,—3),(8,—2,6) and (2,2,—2) lie in a plane.

4. Find the equation of the plane which passes throughthe points (2,—1,6) and (1,-2,4) and is perpendicular to the plane x—2y-®z+9 * 0.

5. Find the equation ef a plane through the threepoints (1,2,3), (—1,—2,—5J), and (4,—2,4). What are its intercepts on the axes?

6» Find the equations of th# line through the. point (2,—1,-3) whose direction eoslnes are proportional to 3, 2, and 7.

7* A l i n e m a k i n g a n g l e s o f 60®, 45®, a n d 60® w i t h t h e X , y , a n d z a x e s r e a p e o t i T e l y p a s s e s t h r o u g h t h e p o i n t (1,—3,1). What a r e i t s e q u a t i o n s ?

8. Find the equation of a plane through the linex+y+z « 0 2x+y+2z**l

and parallel to the plane 3x+2y+3z+2 “ 0.

9. Find the equations of a line passing through the point (1,5,—3) and (2,3,—3).

10. Find the equations of the plane perpendicular to the line joining (2,4,6) and (—6,—4,—2) at i ts middle po in t•

11. Find the equation of the locus of points at a distance +3 from the plane x+y+z+3 * 0.

12. A plane passes through the point (2,6,—7) and is parallel to the plane 3x—4y+12z+17 « 0. Find its equation and distance from the given plane.

13. Find the angle between the linesX - 2y+3 ajnd ^ z * 3y—2 z = 2x+3.


47

14* What is the equation of the plane determined by the point (1 ,3 ,—1) and th e l in e

Ex+Sy—6 * O 3y*7a+l * 0?

15. Find the e^imtion of a pleme passing through the line

x+2y+l » 0 y—3z+4 ■ O

and perpendicular to the plane x+y+z = 0*

16. Find the distance from the lin e3x—Sy+7 * 0 2y+z^ * 0

to the point (4 ,5 ,—6 ).

17. Find the d istance between the l in e s determinedby the points (1 ,2 ,1 ) , (—1 ,2 ,—1 ), and (2 ,1 ,3 ) , (3 ,1 ,2 ) .

18. Find a point in the xy^plane equally d is ta n t from the o rig in and the points (1 ,2 ,—1) and (3 ,—1 ,-3 ) .

19. What two planes are represented by the equation x&+4xy+3y2+xm+yZ+X+5y+2z—2 * 0?

20. What is the equation of the plane determined by the two lin e s

8x—y—13 * 0 _ 7x—3y—22 * 0y+4z+l *0 2x + 325— 5 « Of

21. Show th a t the lin e = ^"^1 is p a ra lle lto the plane aix+biy+c%z+d% * 0 i f , and only i f , aai+bbi+ccx • 0, and perpendicular i f , and only i f

- - è -a i ©I @1

22. Find the equation of the plane through the o rig in and through the in te rse c tio n of th e planes 4%x+B%y++Dq • O and Aox+Bgy+Cgz+Dg “ 0.

23. Find the equations of the planes which b ise c t the angles formed by the planes 2x—y+2z = 0 and x+2y—2z « 6.

24. I f Fi(X]^,y2 ,Z]_) and Fg(x2 ,y2 ,Z2 ) are any two given


48

points, find the loons of a third point Q, such that and PgQ are in a constant ra tio .

Miscellaneous Exercises

1# Find the point in the zy—plane whioh is equally d istant from the origin and the points (1,2,-1) and (3 ,-1 ,-^) •

Zm Find a point in the plane x+y+a * 0 which is eqaallj distant from the points (0,1,0), (1,-2,1) and(2,3,1).

3* Find the distance of the point (4,5,—6) from the line

3x—2y+7 •* 0 2y+z^ * 0.

4. Find the locus of a point which moves so that the sum of in distances from any number of given planes is constant.

5. ) and P2(xg,yg,zg) are any two points. Find the locus of a point Q in the xy plane such that angle PiQPg is a right angle.

6. Show analytically that the locus of points equally distant from any three given points is a straight line perpendicular to the plane of the three points.

7. Prove that the planes, each of whioh is determined by an edge and the middle point of the opposite edge of any tetrahedron meet in a point.

8. Find the volume of the tetrahedron whose faces arethe planes 7x—y—8z-20 ■ 0, 5x—y+z+8 = 0, x+3y—4z+9 = 0,2x—y—Bf—14 » 0.

9. Prove that the straight line ^ is perpendicular to the plane containing the lines2E » 2L « 5, and 5 « Z « ^ is x(m-n)+y(n- )+z( -n) » 0.m n n ^ m

10. Prove that the straight line ^ ^ aX “ 'Sm “ of»tt Z * 5. a l l lie in the same plane i f

Z i “1 «1::^(b—c) + ^ (c -a ) + 2i(a—h) » 0.

m n


49

11. I f a, b, o are the In te rcep ts on the axes o f a plane and i f p i s the perpendicular on the plane from the o rig in then

1 +1 . 1 « 1 22 * ^ * gZ p2‘

12. Prove tha t i f O is the angle between the lin e s

^ . 2 2 1 . 2 2 1 and 2SZ& . 2 2 a . 2 2 am-i

then the sho rtest d istance between these l in e s i s

1sin #

y g -y i

“1

®2r*i

“1

^2


50

Clutpter III

Su7ftto«8 of the Seoond Degree

36» General equation of the seeond degree»— The most

general equation of the second degree Is of the form

d%^+By^+0s*+23yz+2Gzz*2E%y»2nz+2vy+2wz+D » 0 (1)If there Is a real locus» (1) Is the equation of a

eonleoid since every plane section of (1) Is a conic

section#

Although the genereûL equation of the second degree

contains ten constants there are really only nine since

we can divide both members of the equation by any constant

without altering the relation between the variables x, y,

Z»37. Simplest forms of equations of oonlcolds.— The

following equations are the sliQ>lest forms of the equa

tions of conlcolds.

(2) ^ 2 ^ " dps (4> Ax®+By2+D - 0

38. To find the points where a given straight line

cuts the surface represented by the general equation of

the second degree.— Let the egiatlon of the given line


51

be

^frem whioh we bmv# % - y - yi*wr# * “ « ! ♦ » •

Swbetitwilaff bbeae valwe# Q t %, y , m In the genapel

equationA%^*hy^+0%^+2yyz+3G2ac^2a%y+&a%*2ey+2ww+D * 0 (2)

we haven( )^eC ( g^+nr )^*2W ( y ivn r) ( a^<«eir)

v2Q< %i^nr) ( i i* & ) *%m( % itA ) (y% e«y)v2w(zi*^)4^3v(yi*#r) *2w(e%*nr) ♦ n * O (3)

JLrranging th e te rn s in th e l e f t menber o f (3) In deeoeading powers o f r

r^ ( A^^Bi^+Cn*+2ynii^2Gn i^ZEJim) *2r[ { A/&Gn+Bn)%^+( ae»yhi^]^y^* ( Cn+yke&G; zi4^,(#3vm»2wn ] +

v&#yisi*2my^*2Qg^Z]^*2n%^*&vy%*2wÉ]^*D • 0 <3 )

(3 ) i s an eq u a tio n o f th e seoond degree in r . I f the o eastan t term la t h i s eq u a tio n and th e o o e f f io le a ts o f r and r^ a re a l l ae ro , th en th e l i n e l i e s e n t i r e ly in th e ooaieoid# I f i t does no t l i e in th e ooniooid th en It a e e ts th e ooniooid in one, two, o r no po in ts* I f two

ro o ts o f (3 ) are equal hu t oppo site in s ig n , th«o. th e


60

o f r I s aoro* 1%

) ♦a<*atj + /a^ ) ♦HC )

*a/^+vm*#n « 0 (4)

aad. hen## ( 4 ) i e th e e g a a t lo n o f t h e p la n e t h a t h ta e o t#

a l l eh ord a o f t h e o o n io o id who## d l r o o t io n o o o ln e a a r e

/4 ' m$ m#

I f | 3 ) h a s o n ly on# d i a t i n o t r o o t

OA&],^Bsyi+Gnai+r( ) *3( *n Arm

•«m3* — ( A)^**hm?*Ômê0Ame0%L&2Ejüs) (4]^*ehgr2*^»l*+2ïy^âi

e0GS] %) +0mciyie0n%2e2vy%e0»a2eD) * 0 < 8)

Only on# a o lu t i e n f o r r g i v e s th e o o n d it io n f o r ta n —

g e n e y and hanoe ( 5 ) i s t h e e q u a t io n o f th e ta n g e n t p la n e

t o t h e o o n io o id .

3 9 . The eon #a ^ ± ^ ± ^ • 0 . — T ak in g a l l th e p o s

s i b l e co m b in a tio n s o f s i g n s we h ave th e fo n r s u r f a c e s

The l a s t t h r e e o r a t i o n s ea o h h ave two term s w it h

t h e same s i g n and one w it h o p p o s i t e s ig n so we have h u t

tw o d i s t i n c t o a s e s t o d is c u s s *


53Xmii u« examine th e two egaatlon#

UK H - % * 0 (X).h* e^

Iqnn tlon (1) may be w r it te n (=) ♦(%) e (^ ) ■ 0

and elnee th l e e tn a tlo n may be e m tle fle d only f o r the ooordlnatea (0*0,0)* (IV re p ré sen ta th e s in g le p o in t, the o r ig in .

Xqixatlon (3) la a la e a a t l a f le d by (0 ,0 ,0 ) ao th e

o r ig in la a po in t on th e leeua o f (z)m ^ a e I f ^ i ty i t a i la a poin t on th e an rfaee , th en —xi,-flTx!"*l • po in t onthe aurfaee and th e anrfaee la aymmetrloaX w ith reapeet to the o r ig in .

I f fx(%i#yx*a%) la a iy p o in t on th e ao rfaee , the ooordlnatea o f any p o in t on any s t r a ig h t l in e th r o n g the o r ig in and a re mxx, ma^ where m la an a rb i tr a ryfa o to r .

S u b s titu tin g th ese ooordlnatea In (S)

♦ n * > - 0a3 b^ @3

whloh wo know i s tru e sin ce 1# on th e su^-face . Hence th e l in e through th e o r ig in and l i e s


54

o a th e e a r f& e e , end th e e a r fa e e l e t h e r e f o r e

eem poeed o f s t r a i g h t l l a e e pem elng th r o a g h th e o r i g in

end l e t h e r e f o r e & eene#

Sow I f we l e t a • 0 e q n e t lo n ( 2 ) beeom ee

w h leh l e th e e g n e t lo n o f two a t r e i g h t l ln e e *

I f we l e t a " we h ave

ae t h a t t h e e e o t l o n made b y t h e p la n e a * a% ( p a r a l l e l )

t o t h e ry ^ p la n e) i e at>hyperbola*

I f y « 0 we a g a in h a re two e t r a i g h t l l n e e

5 — M • 0 and 5 * & " 0a e a eI n t e r e e o t ln g a t t h e o r i g i n .

I f y y^ we h ave am h y p e rb o la a e t h e p la n e e e e t lo n

and a g n a t io n (S ) beeom ee

n8 yi*is r -

I f % « 0 we h a v e

s - sw h ioh i e a s i n g l e p o in t a t t h e o r ig in *

I f n • we h ave


99

th# pX«n* lorn# are #llip### wMBh Inereaee Inmagnltnt# a# tb# plan# z • x x l a talpm f a r th e r and farther

fuaaar fro » t h # e r l g i a #

If in agnation t « a, than

$ ? ® w h lo h l a t h o o ^ ia a t lo n o f a o l r o n l a r o o n o #

do* Tho o o n tra l gnadrioo#*- 25 X ^ ^w • 1*—T a h ln d o i l p o a a l h l o o o m b in a t lo m a o f w o a i g n a w o h a v o

t h o O q n a tlO n a ,

S * S * S - * i « â > S - 5 - *

S " S " S ' * S " ï » * S " '

I f e a o h o f t h o s e o g n a t lo n a l a a a t l a f l e d h y » i » 7 i » s i

t h e n I t l a a l s o s a t i s f i e d b y o » d t h e o r lg lm

l a t h o e o n t o r o f t h o s e a n r f a o e a a n d t h o a o r f a o o a a r e

o a lX o d o o n t r a l g i ia d r lo a f o r t h i s r e a s o n .

I f l a o q n a t io m ( 1 ) s > 0

a n d t h o p la n e a o o t l o n a m ade b y t h o s y ^ l a n o a r e o l l l p a o a

w i t h s e a l —a x e s a an d b*


56

" *X.

I f < e*, the eee tlo n aede hy the plane e*e% le an e lllp ae* I f # e*, we have a elngle po in t on the 9-axle* I f e^#^ th e eee tlo n le Ime&lnaiy.

Slnee s • glvee the eame re e n lte , the enrfaee le eymmetyleal w ith r e e ^ o t to th e plane n - 0, and le bonnded by the planee s « e, and s * —e* All eeetlone made by planee p a ra l le l to the ny-pXane a re e lllp eee whleh deereaee In magnitude as we tak e z ^ f a r th e r from the xy-plane and f in a l ly reaoh the two p o in ts ( 0 ,0 ,e) and (0 ,0 ,—e ) •

Slnee i , y , z appear aynm etrleally In ofnation (1) we oan make s im ila r statem ents fo r x and y* That i s , the sn rfaee Is eymmetrleal w ith respeo t to th e xe-plane and Is bounded by the planes y » b and y » -b , a l l eeetlone made by planes p a ra l le l to the zx-plane are e ll ip s e s w ith semi-axes a and 2 aa we go fa r th e r out from the sx-^plane we reaoh the p o in ts (0 ,b ,0 ) , (0 ,-b ,0 )*

This aurfaee i s an e l l ip s o id elnee a l l plane aee tlonsare e l l ip s e s .

I f b « e , ^ e • 1 whleh Is th e eqnation o fa* b*


57

th a a X I lp s o lA o f r o T O lt i* t io n o r p r o l a t e a p h o r o lA , fo rm a t

t y ro T O lT la g am a l l i p a a a t o u t i t s m a jo r a x l a z*

I f a • by ^ " 1 w h lo h i s t h o o g iu a t lo a o f th#

o l l l p o o i t o f r e v o l u t i o n o r o b l a t e s p h e r o id fo rm ed b y r e

v o lv in g a n e l l i p s e a b o u t i t s m in o r a x i s a .

I f a ■ b • Oy t h e e q u a t io n i s t h e e q u a t io n o f a

sp h e r e w i t h r a d iu s a .

Sow l e t u s e o n s id o r e q u a t io n ( 2 ) y 3 ^ — ^ " 1#• ^

I f s « Oy ^ ^ * 1 Whleh l a t h e e q u a t io n o f a

h y p e r b o la w i t h i t s t r a n s v e r s e a x i s a lo n g #X*

I f a » * 1 , 2 ^ — ^ " 5 ^ I f 2 * ■ 0 * we h a v e a

p a i r o f i n t e r s e o t i n g l i n e s , 2 + Z - o an d 2 — Z » 0 .. # jj* ® a e

I f s * y o*y £ — C ■ 1 — —1 — i s a h y p e r b o la w i t h tr a n sv e r se a « ^ 0 * * # a S

a x i s p a r a l l e l t o t h e x - a z i s * I f o ^ , ^ — 1

w h io h i s t h e e q u a t io n o f a h y p e r b o la w i t h t r a n s v e r s e a x is

p a r a l l e l t o t h e y - a x l s .

Aa 2 1 in e r e a s e a fro m 0 t o e t h e v e r t i e e s o f t h e h y

p e r b o la oome o l o s e t o g e t h e r t i l l f o r s *• ± e we h a v e tw o

i n t e r s e c t i n g l i n e s .

S in o e z " — g i v e s t h e sam e s e c t i o n s t h e s u r f a c e

i s s y m m e tr io a l w i t h r e s p e o t t o t h e x jM P la n e .

I f a • o , 2 ?^ — ^ • 1 an d t h e h y p e r b o lo id i s a


58

hyperbole la of roTOluàlon*

L o t u s now o o n s i d s r t h e e q o m tio n

I f z " 0 , ^ e • —1 an d t h e s e o t l o n l a Im a g in a r y .

I f X ■

g g ÿI f x ^ " a ^ w e h a v e t h e p o i n t s ( a * 0 , 0 ) a n d ( - a , 0 *0 ) .

I f •• 1 i s a n e l l i p s e w i t h s e m i -

a x e s h a n d j s . I f x^<^ a% t h e s e o t i o n i s im a g in a r y *

I f s • 0 t h e s e o t i o n i s ^ — z £ « 1 id t ie h i s a hy—

p e r h o la w i t h t r a n s v e r s e a x i s a lo n g t h e x - a x i s * I f x

z * a%, ^ ^ » 1 ♦ i*^?irtiioh i s a h y p e r b o la w i t h

t r a n s v e r s e a x i s p a r a l l e l t o t h e x —a x i s *

T h is s u r f a c e i s a b i p a r t e d h y p e r b o lo id * o r a h y

p e r b o l o i d o f tw o s h e e t s h a v in g t h e x - a x i s a s i t s a x i s

a n d c o n s i s t s o f tw o p a r t s w i t h n o p a r t o f t h e h y p e r b o la

b e t w e e n x • a a n d x • - a *

41* The p a r a b o lo id s * ^ * 4 p s . — i n o o n s i d e r -

i n g t h e p a r a b o l o i d we w i l l c o n s i d e r ( 1 ) t h e e l l i p t i c

p a r a b o lo i d Z E + z E • 4 p z a n d ( 2 ) t h e h y p e r b o l i c p a r a b o l—• ft*®

o i d ^ ^ ■ 4 p s* I n b o t h c a s e s l e t p b e p o s i t i v e a n d


59

g r e a t e r th a n 0 $

I n < x) l e t a • 0 . t h e n 2 É + a d . 0 a n d j e e h a r e a

s i n g l e p o in ty t h e o r i g in * I f * “ *x*

w h le h i s t h e e q m t l o n e f an e l X i p e e w i t h se n é F -a x es a

and b . I f s - - S i t h e e e e t l o n i s la a g in a v T f

I n ^ ^ - * w l e t « - e . T hen g - Z | - 0 o r

(|^ — ^ 1 ( 2 + m 0 id i io h i s t h e e g o a t l o n o f tw o i n t e r -

s e o t l n g l i n e s * I f s » s i , ^ » 4 p s i a n d t h e s e e —

t i o n s a r e h y p e r b o la e w i t h t r a n e w e r e e a x e s p a r a l l e l t e

t h e x - e x l e * I f s * —s ^ , 2 ^ — ^ ■ —d p s^ o r ^ ^ ■

4 p s j and th e s e o t i o n s a r e h a ^ e r b o la e w i t h t r a n s v e r s e

a x e s p a r a l l e l t o t h e y^ -ax is*

I f X « x ^ , ^ • 4 p s — sA iie h i s t h e e ^ n a t lo n o f

a p a r a b o la w i t h s a s th e a x i s * When x • O, 3 ^ ■ 4 p s a n d

t h e p a r a b o la h a s I t s v e r t e x a t t h e o r i g i n *

I f 7 • 7 ]^, ^ * 4 p s — w h ic h i s a p a r a b o la w it h

t h e zp -a x ls a s t h e a x i s * When 7 » 0 , 2 ^ » 4 p s »nA t h ea *

p a r a b o la h a s i t s v e r t e x a t t h e o r i g in *

4 6 . C7 l l n d e r s * - The e q u a t io n A x*4-B7 ^+D ■ 0 ( 1 ) , i s

p l a i n l y o f t h e fo r m F ( x , 7 ) » 0 . L e t p b e an7 p o i n t I n

t h e x j ^ l a n e w h o se c o o r d i n a t e s s a t i s f y t h e e q u a t io n


60

F ( x , y ) - ©♦ T hen e v e r y p o i n t o n t h e l i n e th r o u g h P

p a r a l l e l t o t h e a - e x im h a s t h e sam e x a n d y c o o r d i n a t e

a e t h e p o i n t P a n d a l l a u e h p o in t a a r e o n th e a u r f a e e

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n e g a t i v e s i g n h e f o r e X a n d A i s b e t w e e n e ^ a n d b ^ t h o

s u f f a e e i s a h y p e r b o lo id o f o n o s h e e t ; i f À i s b e t w e e n b^

a n d a ^ i t i s a h y p e r b o lo id o f tw o s h e e t s .

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Sxeroiae#

1# T h ree e o n f e o a l o o n io o id # m eet I n a p o i n t , a n d a c e n t r a l p la n # o f eaO h 1 # draw n p a r a l l e l t o i t s t a n g e n t p la n # a t t h a t p o in t # P r o v e t h a t , o n e o f t h e t h r e e s e o t i o n # w i l l b e an e l l i p s e , on e a h y p e r b o la , «MA o n e w i l l b e im a g in a r y #

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70

3« I f th p o q g h & g i v e n e t r a i g h t l i n e t a n g e n t p la n e e are^ A r a m t e a e y e t e m e f e m f a e a l e ^ t h e e o r r e e p o n d — l a g g e n e r a t # a h y p a r b e l l # p a r a h a i e l d .

4$ g h e v t h a t t h e p o i n t e on a e y a te m e f o e n f o e a l e w hlO h a r e e a o h t h a t t h e n o r m a l# a r e p a r a l l e l t o a g i v e n l i n e a r e o n a r e c t a n g u l a r h y p e r b o la .

5* Sheer t h a t t h e l o e u e o f t h e p o l a r o f a g i v e n l i n e w i t h r e a p e e t t o a e y e te m o f o e n f o e a l e l e a h y p e r h e i lo p a r a h e lo lA o n e o f w h o e e a e y iq p t o t lo p la n e e i e p e r p e n - d l e u l a r t o t h e g i v e n l i n e *

6 « r i a n e a a r e draw n a l l p a e e l n g t h r o u g h a f i x e d e t r a i g h t l i n e an d e a o h t o u c h i n g o n e o f a s e t e f oon-* f e e a X o l l l p e o l d e g f i n d t h e lo o u e o f t h e i r p e i n t e o f o o n t a o t *

7 . T h ro u g h a s t r a i g h t l i n e i n o n e e f t h e p r i n e i p a l p l a n e e t a n g e n t p l a n e e a r e d raw n t o a s e r i e s o f o o n f o e a l e l l i p e e i d e ; p r o v e t h a t t h e p o i n t s o f o o n t a o t l i e o n a p l a n e , an d t h a t t h e n o r m a le a t t h e s e p o i n t s p a s s t h r o u g h a f i x e d p o in t *

8 * I f tw o o o n f o e a l s u r f a o e # a r e o u t h y a p l a n e , t h e s e o t i o n s m ade h y t h e p la n e l i e o n a s t r a i g h t l i n e *

8 * 7 , a n d ^ a r e tw o p o i n t s on a g e n e r a t o r o f a h y—p e r h o l o l d : 7 , a n d % t h e o o r r e s p e n d in g p o i n t s on a o o n f o e a l fa r p e r h o lo id * Show t h a t 7 A i s a g e n e r a t o r o f t h e l a t t e r , a n d t h a t 7Q ** P Q*

1 0 * I f t h r e e o f t h e g e n e r a t i n g l i e s o f th e e n v o i- , o p in g o o n e o f a p a r a b o l o i d a r e m u t u a l ly p e r p e n d ie u — l a r , sh o w t h a t t h e v e r t e x w i l l h e o n a p a r a b o lo id , a n d t h a t t h e p o l a r p la n e o f t h e v e r t e x w i l l a lw a y s t o u o h a n o t h e r p a r a b o lo id *

1 1 * A t a g i v e n p o i n t 0 t h e t a n g e n t p l a n e s t o t h e t h r e e o o n i o o i d s w h ic h p a s s t h r o u g h 0 , an d a r e o o n f o e a l w i t h a g i v e n o o n i o o i d , a r e d ra w n ; show t h a t t h e s e t a n g e n t p l a n e s a n d t h e p o l a r p la n e o f 0 fo r m a t e t r a h e d r o n w h lo h i s s e l f —c o n j u g a t e w i t h r e s p e o t t o t h e g i v e n o o n io o id *


n

th é o«xif e f eeesK.#«14 #r# $h# M### #E&y$#%b#WL& mfk q##L.;#hm## ##W k ####»# %W«m#hp &## T#y%#z.

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C h a p t e r I T

T h e S p h e r e

5 2 * T h e e g i o a t i o n o f t h e s p h e r e w h o s e e e n t e r I s

en A w h o s e r a d i u s I s r f o l l o w s d l r e o t l y

f r o m t h e f o r m u l a f o r t h e d l s t a n e e h e t w e e n tw o p o i n t s

i n e p a e e ,

(%eu)^+(y+T)^4^(z4w)^ ■ r^f ' *a n d t h e e q u a t i o n o f a n y s p h e r e i s o f t h e fo r m

%^+y^^a2*2uz+2vy+2wz+D » 0 (1)

w h e r e r • •✓«^-►▼S+wS—p*

S i n e e t h e g e n e r a l e q u a t i o n o f t h e s p h e r e o o n *

t a i n e f o u r o o n s t a W s w e o a n f i n d t h e e q u a t i o n o f t h e

s p h e r e w h e n w e h a v e f o u r c o n d i t i o n s g i v e n . F o r e x

a m p le » w e o a n f i n d t h e e q u a t i o n o f a s p h e r e w h io h

p a s s e s t h r o u g h a n y f o u r p o i n t s *

I f t h e c e n t e r o f t h e s p h e r e i s a t t h e o r i g i n t h e

e q u a t i o n r e d u c e s t o

x^+y^e*^ • r^ (2)

I f i s n e g a t i v e t h e l o c u s i s im a g in a r y *

I f ( x » y » s ) i s a p o i n t o n ( 2 ) t h e n ( - x , - y » - a ) i s a l s o a

p o i n t o n ( 2 ) a n d t h e s p h e r e i s s y m m e t r i o a l w i t h r e s p e o t

t o t h e t h r e e a x e s *


73

5 3 * D e f i n i t i o n o f t a n g e n b p l a n e t o a e n r f a o e . —

Two l i n e # , ta n g p e n t t o a s n r f a e e a t a p o i n t , A e t e r m ln « a

p l a n e w h l o h I s c a l l e d t h e t a n g e n t p l a n e t o t h e s u r f a e a

a t t h a t p o i n t . T h e p o i n t 1 # e a Q lle d t h e p o i n t o f ta j> -

g e n o y o r p o i n t o f o o n t a o t .

A l l t h e t a n g e n t l i n o # d r a w n t o a a n r f a o e a t a n y

p o i n t o n I t l i e I n a p l a n e . T h e p r o o f o f t h l # s ta te n a n t

1 # l e f t t o t h e s t u d e n t .

5 4 . T a in g e n t p l a n e t o a s p h e r e S i n c e t h # s h o r te s t

d l s t a n c e f r o m a p o i n t t o a p l a n e 1 # t h e p e r p e n d l o u l a r

f r o m t h e p o i n t t o t h e p l a n e . I t f o l l o w # t h a t I f a p lane

1 # p e r p e n d l o u l a r t o a r a d i u s o f a s p h e r e a t t h e p o i n t la

w h l o h I t m e e t s t h e s p h e r e , t h e n e v e r y o t h e r p o i n t o f the

p l a n e I s e x t e r i o r t o t h e s p h e r e a n d t h e p l a n e I s o a lla &

a t a n g e n t p l a n e t o t h e s p h e r e .

l e t t h e p o i n t ( x ^ . y i . a ^ ) h e a p o i n t o n t h e s p h e r e

x^-«^y^-e>s^e2 tiX'»>2 vy<e>2 wze>D « O ( l l

T h e n t h e p l a n e p e r p e n d i c u l a r t o t h e r a d i u s w h le h

m e e t s t h e s p h e r e i n t h e p o i n t 3 E ^ » y x » * x

( x x * u ) ( » ^ ] ^ ) * ( y 2 e v ) ( y - y ] ^ ) * ( z ] ^ + w ) ( a —s ^ ) - O ( 2 )


74

s i n e e t h e d l r e o t l o n o o s l n e s o f t h e l i n e J o in in g t h e

p o i n t s an d t , - w ) a r e p r o p o r t i o n a l t o

Prom e q u a t io n ( 2 ) we o b t a i n

:ai%+yiy*%i»+n%+vy+ws-ac2^1^1^""n%]r^l"^l " 0 O )

But ainoe H as on the sphere

H en ee a d d in g a n d a n h t r a c t i n g n% % +vyi*wai t o t h e

l e f t member o f ( 3 ) g i v e s

XiXvj]^y-#-»^a-«iCx+Xi)>T(y+yi)+w(a^ai) ♦ P • 0

w h io h i s t h e e q u a t io n o f t h e p la n e t a n g e n t to t h e s p h e r e

a t t h e p o i n t X i ,y x » ® l*

55# l i n e t a n g e n t t o a s p h e r e # — E v e r y l i n e i n t h e

t a n g e n t p la n e t o a s p h e r e an d p a s s i n g th r o u g h t h e p o i n t

o f t a n g e n e y m e e t s t h e s p h e r e i n o n l y o n e p o i n t and i s a

t a n g e n t l i n e t o t h e s p h e r e # C l e a r l y a t a n g e n t l i n e i s

p e r p e n d i e n la r t o t h e r a d iu s d raw n t o t h e p o i n t o f t a n -

g e n s y #

S i n c e t h e t a n g e n t p la n e a t a p o i n t i s t h e l o c u s o f

a l l t a n g e n t l i n e s a t t h a t p o i n t , i t I s c l e a r t h a t tw o

o f t h e t a n g e n t l i n e s a t a p o in t d e te r m in e t h e t a n g e n t

p la n e a t t h a t p o in t #


76

5 6 . L e n g th ot t a n g e n t l i n e fr o m a p o i n t t o a

a p h e r e . - A l i n e t a n g e n t t o a a p h e r e i s p e r p e n d ic u la r

t o t h e r a d iu s draw n t o t h e p o i n t o f c o n t a c t , an d

h e n c e t h e l e n g t h o f t h e t a n g e n t t o a s p h e r e fr o m an

e x t e r i o r p o in t may h e fo u n d b y m eans o f a r i ^ t t r i

a n g l e . X>et P x ( z i » y x f Z i ) b e th e p o in t e x t e r i o r t o th e

s p h e r e

x*+y&+s&e2u%e2wy*2wne IM0 ( 1 )

Then t h e s c u a r e o f t h e T^, ,,/ / q / /

d i s t a n c e fro m bo th e c e n t e r o f ^

t h e s p h e r e i s

(x ^ + u )* + fy x + T )2 ^ (a ^ + w )* and t h e

sq u a r e o f t h e r a d iu s o f t h e s p h e r e i s u^+v^+w^—D . Henee

t h e sq u a r e o f t h e l e n g t h o f t h e t a n g e n t i s

(x2+u)^+(yX»T)^+(sg^*w)B-(u2+w^+w%-D)

o r Z x ^ + y x ^ + si^ * 2 tkX2 +2 T yx» 2 wzg^+D.

T hat i s , i f t h e c o o r d i n a t e s o f a p o i n t e x t e r i o r to

a s p h e r e a r e s u b s t i t u t e d i n t h e l e f t member o f e q u a t io n

( 1 ) t h e r e s u l t i s t h e s q u a r e o f t h e l e n g t h o f t h e t a n

g e n t from t h a t p o in t t o t h e s p h e r e .


76

57* I n t o r s ^ o t i o n o f tw o o ^ h o r o a .— L o t t h e oquar-

t l o n s o f th o s p h e r e s b e

and %^4^y&*s^4^2ti2i+2T23r+2#g#*Dg ■ 0 ( 8 g " 0 ) .

Them ®2 r S i « 0 I s

2(mg^l)z+2(V g r^ )y 4 ^ 2 (> s+ D g -D i - 0

w h io h I s t h e e q u a t io n o f a p la n e r e p r e s e n t i n g a l l

p o i n t s common t o b o t h s p h e r e s . T h e r e f o r e a l l p o i n t s

common t o tw o i n t e r s e c t i n g s p h e r e s l i e o n a p la n e #

5 8 . The r a d i c a l p la n e o f tw o s p h e r e s .— The equa

t i o n • 0 r e p r e s e n t s a p la n e w h e th e r t h e tw o

s p h e r e s i n t e r s e c t o r n o t a n d t h i s p la n e i s c a l l e d t h e

r a d i c a l p la n e o f t h e tw o s p h e r e s . The r a d io e d p la n e o f

tw o s p h e r e s i s u n iq u e i n t h a t t a n g e n t s draw n t o t h e two

s p h e r e s fr o m a n y p o i n t i n t h i s r a d i c a l p la n e a r e e q u a l .

L e t b e a n y p o i n t i n t h e r a d i c a l p la n e

o f t h e tw o s p h e r e s . T hen t h e l e n g t h s o f t h e t a n g e n t s

fr o m t h i s p o i n t a r e

xq^+yX^+Sj^^+SuiXi+STiyi-i-SwQ^Zi Lj

an d

anA t h e d i f f e r e n c e b e tw e e n t h e i r l e n g t h s i s


?7

t u t t h e p o ixrt H w o n t h e r a d l o e l p la n e aaA

h e n e e t h e d i f f e r e n c e o f t h e l e n g t h # o f t h e t a n g e n t s I s

s e r e #

I b c e r e l s e s

1 # f i n d t h e e q im t io n o f t h e p la n e t a n g e n t t o t h e s p h e r e x g ^ y g » s 8 - Ox 6y » 2 g - 8 1 « 0 a t t h e p o i n t ( 3 , 0 , 2 ) #

2 » f i n d t h e e^ r o a tlen o f t h e s p h e r e s h l o h h a s t h e c e n t e r ( 3 , 0 , - 2 ) an d p a s s e s th r o n g h ( 1 , 6 , —6 )#

3 # 7 ln d t h e e q u a t io n o f t h e s p h e r e w h ic h p a e s e s th r c n g ht h e p o i n t s ( 0 , 0 , 0 ) , ( 0 , 2 , 0 ) , ( 4 , 0 , 0 ) , an d 1 0 , 0 , - ^ ) #

4# F in d t h e e q u a t io n o f t h e s p h e r e w h ic h h a s I t s c e n t e ron t h e p ^ -o z ls and p a s s e s th r o u g h t h e p o i n t s ( 0 , 2 , 2 ) and ( 4 , 0 , 0 ) #

9# F in d t h e l e n g t h o f a t a n g e n t from t h e p o i n t ( 9 , 1 0 , 1 4 ) t o t h e s p h e r e (% + 2 )2 + (y * 3 )2 + (g —4 ) 2 ■ 29#

6 # F in d t h e e q u a t io n o f t h e s p h e r e w h ic h h a s t h e l i n ej o i n i n g ( 4 , —4 , 5 ) and ( 2 , 0 , 2 ) a s a d ia m e te r #

99# R a d ic a l c e n t e r o f f o u r s p h e r e s # — L e t t h e f o u r

s p h e r e s h e " 0 , 8 g » 0 , 3 ^ • 0 , 8 4 • 0 an d l e t n o

t h r e e o f t h e i r c e n t e r s h e o o l l l n e a r and t h e f o u r c e n t e r s

n o t c o p la n a r # The r a d i c a l p la n e s o f t h e s p h e r e s t a k e n

I n p a i r s a r e

^1"®2 " 0 (1) Sg-6g « 0 (2) 82-64 - 0 (3)

Si-63 - 0 (4 ) 82-84 - 0 (5) 83-8^ - 0 (6)


78

P la a # » ( 1 > , ( 2 ) , ( 3 ) m e e t l a o a # p o l a t , (x^»3Tx»

»%)# a i a a e ( 3 ) — ( 1 ) g lT #m ( S ) a l l p o l a t a eonm oa t o ( 3 )

anA (X ) l i e o a ( 8 )* T h a t l a , ( 3 ) p a a a o a t h r o a t C ^ i» 7 it

S i m i l a r l y ( 4 ) a a d ( 6 ) p a s s t h r o u g h (z i# y x # a % ) a a d

w # h a r o t h o s i x r a d l o a l p la a o a o f f o u r s p h o r o s m oetlagp

l a m p o l a t o a l l o d t h o r a d l o a l o o a t o r o f t h o a p h o r o o .

S x o r e la o a

1 « n m d t h o r a d l o a l p ia n o o f t h o tw o a p h o r o a x^ + y^ *#^ - 6 x * 4 y - 6 a —1 4 • 0 , xZ + y2*a8+ a% -2y+ 8n + 10 * 0 .

2 * T ih d t h o r a d l o a l o o a t o r o f t h o f o u r a p h o r o a x ^ -fj^• 0 , x 2 ^ y 2 ^ » 8 * 2 5 , x 9 + y 2 + a 2 ^ % —4 y + 8 a

—1 4 • 0 , x ® * y 8 * e S —1 4 x * 8 y * 4 o - 6 ■ O*

3# F in d a r o l a t l o B am ong t h o o o h a t a a t s l a t h o o q u a - t lo a a a^+y24# 2 i 2 nx+&oy4 2 wa+d « 0 , x&*y2+m&$^2uxx+2Tiy+ 2 wia*d% • 0 w h io h w i l l m ake t h o r a d i c a l p la a o o f th o tw o a p h o r o a t a n g e a t t o b o t h a p h o r o a .

6 0 . S y a to m o f a p h o r o a t h r o u g h t h o l a t o r a e o t l o a o f

tw o a p h o r o a .— L o t th o o g u a t lo n a o f t h o a p h o r o a b e

x^+y^+aê2uX4^2ry+2wa+D • O (1)

*«a x^+yêz^*2Ux%e2Txye2w2a+])X» O (2).

T h en x^ ey& *a^ + 2u x*2vy*2w a+ D + k( x^ +y^â^+Su^x+ST xy

e2wxa*l>x) " 0 (3)

e o a t a l a a a l l p o i n t a id x lo h a r o oommoa t o t h e tw o a p h o r o a .


79

Egum tlom ( 3 ) r e p r é s e n t e a s p h e r e t m l e s a k • —1

b e e a u s e we may w r i t e t h e e x p ia t io n I n t h e fo r m

( lek )z^e( lek )y^e( lek)**+2(u*kai >%*&( ▼♦kWi)y+2( wekw^)

eDekD] * 0

an d d i v i d i n g thron& h b y l e k , k

w h le h i s th e e q ;a a tlo n o f a ip h e r e * I f k * —1 w e h a v e

t h e e q u a t io n o f a p la n e id i l e h I s t h e r a d i a a l p la n e o f th e

tw o s p h e r e s .

6 1 . O r th o g o n a l s p h e r e s .— Two s p h e r e s a r e s a i d t o

b e o r th o g o n a l ( a t r l ^ t a n g l e s t o e a o h o t h e r ) a t a p o in t

o f I n t e r s e e t l o n I f t h e t a n g e n t p l a n e s t o t h e s p h e r e s a t

t h i s p o in t a r e a t r i g h t a n g l e s t o e a o h o th e r * S i n e s a

p la n e t a n g e n t t o a s p h e r e a t a p o i n t I s p e r p e n d le n la r t o

t h e r a d iu s draw n t o t h a t p o in t I t f o l l o w s t h a t tw o sp h e r e s

a r e o r t h o g o n a l a t a p o i n t I f t h e r a d i i draw n t o t h a t p o in t

a r e p e r p e n d le o la r t o e a o h o t h e r .

l i e t t h e r a d i i o f th e tw o o r t h o g o n a l s p h e r e s b e r^

r g and th e d l s t a n o e b e tw e e n t h e i r c e n t e r s d . T hen d%» r^*

v r £ * an d we t h e r e f o r e h a v e t h a t I f tw o r a d i i draw n t o a

common p o in t on tw o s p h e r e s a r e a t r i g h t a n g l e s t o e a o h


80

t h e n t h e r a d i i draw n t o a l l e n e h p o i n t s a r e a t r i g h t

a n g l e s an d h e n e e I f tw o s p h e r e s a r e o r t h o g o n a l a t o n e

p o i n t o f I n t e r s e e t l o n , t h e y a r e o r t h o g o n a l a t a l l p o i n t s

o f I n t e r s e e t l o n # S n e h s p h e r e s a r e s a i d t o h e o r t h o g o n a l

t o e a o h o t h e r #

62# P o l e a n d p o l a r p la n e # — We h a v e s e e n 5 4 t h a t

t h e e q n a t l o n

x s i e y 3 ; y e s i s e a ( x 4 * x i ) e T ( y e y i ) + w ( * e s i ) e P « O ( 1 )

I s t h e e q i i a t l o n o f t h e t a n g e n t p la n e t o t h e s p h e r e

z3+y&+m2e2n%e2eye2ws+P • 0

a t t h e p o i n t (% i# y i# % i)*

F o r b r e v i t y l e t n s d e n o t e t h e p o i n t h y P^# t h e

p la n e h y p^# a n d t h e s p h e r e b y 8 # S o m a t t e r w h e re P^ I s

t h e p la n e p^ i s d e f i n e d a s t h e p o l a r p la n e o f P^ w i t h

r e s p e o t t o S , a n d t h e p o i n t p]^ i s t h e p o l e o f t h e p la n e

p 2 w i t h r e s p e o t t o 3#

I f t h e p o i n t P^ I s o n t h e s p h e r e t h e p o l a r p la n e

o f t h e p o i n t P^ w i t h r e s p e o t t o 3 i s t h e t a n g e n t p la n e

t o t h e s p h e r e a t t h e p o i n t P^*

43# P o l a r p la n e o f e x t e r i o r p o in t # — L e t t h e p o i n t

^ l C * l » y i » * l ^ * p o i n t e x t e r i o r t o t h e s p h e r e S • O I s


81

thmn b y d é f i n i t i o n , th e p o l a r p la n e o f t h e p o i n t w ith

r e a p e e t t o t h e s p h e r e 3 • 0 i s

I ,e t P2 ( x 2 , 7 2 ,S 2 ) b e a p o i n t on t h e s p h e r e , t h e ta n

g e n t p la n e a t w h io h p a s s e s th r o n g h P]^* The e q m t i o n o f

t h i s t a n g e n t p la n e i s

%g%+ygy+S2Z+n(zg+z) +w(yg*y)*#( zg*s ) *P • 0 (.2)

B n t s in o e p la n e ( 2 ) p a s s e s t h r o u g h P^

% ]^% g^yiyg+s^sg+n(Z]^*% g)+w(yi+yg)4W (zi+% g)+D - 0 (3)

I t f o l l o w s fro m ( 3 ) t h a t t h e p o i n t x g , y g , s g l i e s on th e

p o la r p la n e o f P] s i n o e X g ,y g , s g s a t i s f i e s e q m t i o n ( 1 ) .

H enee we h a v e t h e g e n e r a l s t a t e m e n t

I f t h e p o la r p la n e o f P^ p a s s e s th r o u g h P g , th e n

t h e p o l a r p la n e o f P g p a s s e s th r o n g h P^*

64# The p o l e o f a g i v e n p la n e # — We may f i n d t h e

p o le o f a g iv e n p la n e w i t h r e s p e o t t o a g i v e n s p h e r e a s

f o l l o w s •

L e t t h e e q u a t io n s o f t h e s p h e r e an d t h e p la n e b e

x^+y^+s^+2uz*2vy+2wz+D « 0 ( i)

an d Ax+ By + Cz + D* » 0 ( 2 )

r e s p e o t i v e l y , and l e t h e t h e p o l e o f ( 2 )

w i t h r e s p e o t t o ( 1 ) # Then e q u a t io n ( 8 ) i s e q u i v a l e n t to


82

ov ( g%*w)z^wc^*vy]^*wz2* D » 0

a a d t& te r a fe r e

à . • . & . yi+v» ïo^vÿ% *#m ^ F îEg%Tyy^+wz^'

C _ *i*w»

T h # # e e q ^ a a tlo a a a r a e g a l v a l e n t t o

( jb»^)% i^*Avy^^2 wa2 + d ]H A i^ • 0

Baz^e ( Btnf)y]^+Bw#l*B3WV2 ■ 0

C t e x e C v y i ^ ( ( ^ B ' ) * l ^ a ) ^ - 0 .S o l v i n g t h e e e e q n a t l o a f f o r x , z , we haere t h e p o l e

o f t h e g i v e n p la n e *

S z e r e i a e s

d r o n p A

1 # F in d t h e e q n a t lo m o f t h e a p h e r e w h io h p a s a e a t h r o n g h t h e p o i n t s ( 1 , 1 , 0 ) , ( 0 , 1 , 1 ) , a n l ( 1 , 0 , 1 ) an d w h o se r a d iu s i s 1 1 *

2 . F in d t h e o e n t e r an d r a d iu s o f e a o h o f t h e f o l l o w i n g s p h e r e s »

( a ) x5**y*+*?"3d • 0(h i xfearB4-*5-^+4z • o(e) » 0( d ) xB^y«4-»*—t e ^ 6 y-f8 s —T » 0

3* F in d t h e e q u a t io n o f t h e t a n g e n t p la n e a t t h e p o i n t ( 3 , 9 , 7 ) t o t h e a p h e r e t h r o u g h t h a t p o i n t ir iioae o e n t e r i s a t t h e o r i g i n *


83

4 * 4 M h e r # who## o e n t e r 1 # ( 3 , 0 , - 8 ) p a e s e s th r o u g h th ep o in t ( 5 , 7 . 3 ) . ? i a 4 t h e e t n a t t e n o f t h e t a n g e n t t o th e s p h e r e a t t h i s p o in t *

5 * Tlnd t h e l e n g t h o f t h e t a n g e n t from t h e p o i n t ( 1 3 , 9 , 0 ) t o t h e s p h e r e e8%-#4ye8 e —0 .0 • 0 .

5> F in d t h e r a d l o a l p la n e o f t h e tw o s p h e r e s -6ye2e-15 • 0, z^eya+m^-5ye4n-6 « 0 .

7 . F in d t h e w a lu e e f It I f t h e s p h e r e s z ^ + y ^ e a & -e z -1 0 y <*>1 0 se lc « 0 , % Z *y 8 + z8 + 4 x -4 y * 8 n -1 4 » 0 a r e o r t h o g o n a l t o e a o h o th e r *

8 . F in d t h e lo o n s o f t h e o e n t e r o f a l l s n h e r e s o r th o g o n a l t o t h e s p h e r e s z # + y 3 + s2 —4ye8m -^ • 0 , xe^|r2e*®-4JfaD—1 4 iy ^ « 0 .

9* F in d t h e p o la r p l§ h e o f t h e ' p o i n t ( 3 , 0 , —4 ) w i t h re sp eo t^ *t o t h e s p h e r e % ^*F^+s«-3oD-4ye7s—1 1 • 0* D oes t h i s p la n e to u o h t h e sp h e r e P

10# F in d t h e p o l e o f t h e p la n e & o -3 y + 5 s • 6 w i t h r e s p e o t t o t h e a p h e r e • 1 6 .

1 1 . A s p h e r e h a s i t s o e n t e r a t t h e p o i n t ( 2 , —1 , 0 ) an d i s o r t h o g o n a l t o t h e tw o s p h e r e s z5 + y 2 es& —4 y * 8 s —8 • 0 , z 2 + ^ 8 e * 2 - .i2 x -1 4 a r ^ • 0 . F in d I t s é q u a t io n .

1 2 . F i ^ t h e d i s t a n o e fr o m t h e o e n t e r o f t h e s p h e r e3C»^yg» « 2 6 m 4 y e 2 * - 1 2 • 0 t o t h e p o l a r p la n e o f t h e p o in t( 3 , 7 , 9 ) w i t h r e s p e o t t o t h i s sp h e r e #

G roup B

1# F in d t h e r a d l o a l p la n e o f t h e tw o s p h e r e s —3x-5y—12 • 0, z«4y»es2.-3ye5»-9 • 0.

2 # F in d t h e r a d l o a l c e n t e r o f t h e f o u r s p h e r e s x ^ 4y®eà® -8 X -4 y e 8 * ^ 1 6 - ^ 0 , x « + y 2 + s 2 4 t o - 4y + 2 » - l l • 0 , x 2 e y 8 ^ s 2 -3 y —z—4 • 0, xP+y«es&-xe2y*4s • 0.

3 . F in d t h e o e n t e r an d r a d iu s o f e a o h o f t h e f o l l o w i n g s p h e r e s .


84

® ) 3t5^y5>*5"'®»-33r^6»-ll • I4) z"+y**#3-%*gyLy»-14 - #

4» A m pb#r# whomp# o e n t e r 1 # ( 0 , 1 , 5 ) p e e e e e th r o u g h t h e p o i n t ( 4 ,« " 1 ,1 ) « 7 1 b A t h é e q o A t lo a o f t h e t e n g e n t t o t h e s p h e r e a t t h i s p o i n t .

5* F in d t h e l e n g t h e f t h e t à n g é n t fro m t h e p o i n t ( 1 0 , 5 , 4 ) t o t h e s p h e r e * * * y 8 * 8 * ^ x > 83^-4 e - 1 0 • 0 .

4 # F in d t h e r a & i e s l n lm n e o f t h e « p h e r e s x * e y 8 e # 8 --1 0 x >6y 4m"5 * 0, %B4y8+n#e@ze7y ëm- • 0.

7* F in d t h e p o l a r p la n e e f t h e p h in t ( 8 : 3 , —5 ) v i t h r e s p e e t t e t h e s p h e r e z # e y 8 * s & -g % -^ * e m 8 0 - 0 » p e e s t h i s p la n e t o n e h t h e s p h e r e ?

8 * F in d t h e p o l e o f t h e p la n e x - 7 y + 8 s « 4 w i t h r e s p e e t t o t h e s p h e r e z h + y 8 + s 8 * 8 5 *

9* F in d t h e v a lu e o f h i f t h e s p h e r e s z ^ + y 8 +s% -3 ] B - ^ e5m —1 2 « 0 , %8 +y&+s&6#S i E » 0 a r e o r t h o g o n a l t o e a o h o t h e r #

1 0 # A s p h e r e h a s i t s o e n t e r a t t h e p o i n t ( 3 , 0 , —4 ) a n d i s o r t h o g o n a l t o t h e tw o s p h e r e s z ^ e 3r8 4 >s8 - 4 z<i>2 a —1 0 * 0 , x 8 e y 8 * s 8 - @ y * 4 » ^ • 0# F in d t h e r a d i u s o f t h e s p h e r e #

1 1 . F in d t h e p o l e o f t h e p la n e t h r o u g h t h e p o i n t s ( 1 , 0 , 2 ) ( 3 , 0 , - ^ ) , ( 1 , 5 ; $ ) w i t h r e s p e e t t o t h e s p h e r e —5 o w y e 3 m -6 » 0 w i t h o u t f i n d i n g t h e e q .u a t io n o f t h e p la n e t h r o u g h t h e s e p o i n t s .

1 8 . F in d t h o d i s t a n e e fr o m t h e o e n t e r o f t h e s p h e r e x $ + y 8 ♦ n g —lQ x e Oy ■4 « e l 4 « 0 t o t h e p o l a r p la n e o f t h e p o i n t ( 1 , 5 , 3 ) w i t h r e s p e o t t o t h e s p h e r e #

G roup C

1 . F in d t h e e q i ia t io n o f t h e s p h e r e w i t h o e n t e r ( 4 , —1 , 2 ) a n d r a d i u s 4#


86

2« F in d t k e e q u a t io n o f t h e s p h e r e w i t h t h e p o i n t s ( 4 , 2 , 1 ) and ( 3 , —7 , 4 ) a s e n d s o f a d ia m e t e r ,

3 , F in d t h e e q u a t io n o f t h e s p h e r e t a n g e n t t o t h e e o o r d ln a t e p la n e s w i t h r a d i n s a ,

4 , F in d t h e o e n t e r a n d r a d iu s o f e a o h o f t h e f o l l o w i n g s p h e r e s ,

( a ) x 8 ^ .y 2 * 8 8 -3 x * 5 y -6 s -y L - 0(h ) * 2 ^ y 8 * e 2 H $ tix * 2 e » -* » -e " • 0( o ) ,qp .y Ig m 0

5 , F in d t h e o e n t e r and r a d iu s o f th e s p h e r e t h a t I s t h e l o o n s o f t h e p o i n t s t h r s s t i m e s a s f a r f r e m t h e p o in t ( a , b , e ) a s fro m t h e o r i g i n *

6 , P o e s t h e l i n e j o i n i n g t h e p o i n t s ( 2 , ^ 1 , ^ ) , (—1 , 2 , 3 ) i n t e r s e e t t h e s p h e r e % 2 + y 2 ^ 2 « lOT F in d t h e p o i n t s o f i n t e r s e c t i o n *

7 , F in d t h e e q u a t io n o f t h e p la n e t a n g e n t t o t h e sp h ere x 2 ear2 ^»2 ^2 m-^3r^8 —1 • 0 a t ( 0 , 1 , —3 ) ,

8 , F in d t h e r a d iu s o f t h e o i r o l e i n w h io h t h e p o la r p la n e o f t h e p o i n t ( 4 , 3 , —1 ) w i t h r e s p e o t t o t h e s p h e r e xS + y2^ »2 m 1 6 o u t s t h e s p h e r e ,

9 , Show t h a t t h e p la n e 3 x * y - 4 * - 1 9 i s t a n g e n t t o th e s p h e r e x 8 * y 2 * s 2 - 8 x - 4 y ^ s —1 2 « 0 and f i n d t h e p o i n t o f o o n t a o t ,

1 0 , F in d t h e s p h e r e th r o u g h t h e o r i g ^ w h io h p a s s e s through t h e p i t e r s e o t i o n o f t h e s p h e r e s x * » F ^ s 2 - a x + 4 y ^ s m 8 • 0 , %2+y2*z2-2x+r-w-10 - 0*

1 1 , F in d t h e e q u a t io n s o f t h e r a d l o a l a x i s o f t h e sp h e r e s x 2 * y 2 * z 3 - 3 3 D - 5 y - e —4 « 0 , x 9 e y 2 4 > z 2 e S x - 3 y - 2 B - 8 « 0 , x 2 + y 2 ♦*3—16 ■ 0,

1 2 , F in d t h e r a d i c a l o e n t e r o f t h e s p h e r e s x ^ + y 2 + z2 -6 x ^ 2 y "-Z+6 - 0 , x 2 + y 2 + z2 —1 0 * 0, x3^y24>z24.2x-33r-15z—6 « 0 ,x2^y2^s3-2xe>4pwi2 « o, *


86

M l# 0 # llm n eou m S z e r o i s e s

1 « Show t h a t t h # « g m t l û n A( a f ) * 8 u z 4 8 v y + 2 wm*D - 0I n w h io h D 1 # a v a r l a b l o , r o p r o s o n t a a f a m i l y o f o o n o o o -t l r o a p h o r o a #

2 # S h e # t h a t t h o lo o n a o f p o i n t a , t h o r a t i o o f w h oao A la t a n o o # from tw o g i r o a p o i n t a l a o o n a t a n t , l a a a p h o ro o x o o p t whom t h o r a t i o l a im l t y #

3 # Show t h a t t h o r a d lo a X p la a o o f tw o a p h o r o a l a p o r p o a - d i o a l a r t o t h o l i n o j o i n i n g t h o l r o o n to r a #

4 # Show t h a t t h o tw o a p h o r o a x 2 + y 2 + g S + a ^ ^ .b iy * O iz o d 2 « 0 , %hoy2 4 ^w**#axoh2 y * o ga+ & g « 0 a r o o r t h o g o n a l w hoa

5# Show t h a t a a p h o r o w h io h o t ita t h o tw o a p h o r o a 3 * 0a a d Sn ■ O o r t h o g o n a l l y , w i l l o u t k i S + k ^ i * 0 o r t h o g o n a l l y #

6 # Show t h a t t h o lo o n a o f m p o i n t a n o h t h a t th o atun o f t h o a ^ a r o a o f w hoao d la t a n e o a fr o m a f i x e d p o i n t a l a o o n a t a n t l a a a p h e r e # "

7* Shew t h a t I f t h e o o o r d ln a t e a o f t h e o x t r e m l t l e a o f w d ia m e t e r a r e ( x n , p ^ , s ^ ) an d t h e n t h e e q m t d o no f t h o a p h o r o may h e W r i t t e n

( M E ^ ) ( x - x g ) ♦ ( ] M 7 2 ) ( y - y g ) ♦ ( z - a ^ ) ( » - z g ) - 0 .

8 # F in d t h e e q u a t io n o f t h e s p h e r e w h io h t o u e h e a t h e p ia n o y * 0 o u t a t h e p la n e z « 0 I n t h e o i r o l e(x -a )2 + (y w _ b )2 m r 2 #

9# A a t r a l g h t l i n o l a d raw n t h r o u g h t h o p o i n t 0 , m e e t in g a f i x e d p ia n o l a Q, a n d o n t h l a l i n o a p o i n t p l a t a k e n a n o h t h a t G P «o4 i a o o n a t a n t . Show t h a t t h e lo o n a o f P l a a s p h e r e p a a a ln g t h r o n g h 0 , w h o se o e n t e r l a on t h e l i n o t h r o u g h 0 p e r p o n d lo u la r t o t h e p la n e #

1 0 . G iv e n t h e a p h e r e x2+y2+% 2 * 2 wz4 2 v y + 2 w z+k * O, f i n d t h o l o o n a o f a l l p o i n t # a n o h t h a t t h o t a n g e n t s fr o m t h e p o i n t s t o t h l a s p h e r e a r e e q u a l I n l e n g t h t o th o r a d lu # o f t h o a p h e r e #


87

11# I f A aad B a r e tw o f i x e d p o i n t a and B m oTes a o t h a t PA “ nPB . p r o v e t h a t th e lo o n a o f P l a a a p h e r e p r o v id e d a ■ 1# P r o v e t h a t a l l e a o h a p h o r o a f o r d i f f e r e n t v a lu e a o f n h a v e a oooraon r a d l o a l p la n e #

12* P r o v e t h a t t h e lo o n a o f a p o i n t , t h e anm o f t h e a q n a r e a o f w hoao d la t a a e e a fr o m t h e a l z f a e e a o f a ehbe l a o o n a ta n t l a a sp h e r e #


68

O hm #ter v

T ra m # f0 rm % tlon o t C oovA ixuittcs : Sym tem s o f C o o r d ln a to s

6 3 * T r a n s l a t i o n o f #%##*— I f wo w is h t o t r a n s l a t e

t h e a x e s t o a new o r i g i n 0 b y m o v in g t h e a x e s

p a r a l l e l t o t h e m s e l v e s we h a v e t h e e g n a t I o n s f o r t r a n s l a

t i o n

X • x"^eh, y • k* 9 £ 9

L e t x , y , s b e t h e o o o r d l n a t e s

o f a p o i n t P r e f e r r e d t o t h e o l d a x e s

a n d ( x \ y ^ t h e c o o r d i n a t e s o f a/ /

p o i n t r e f e r r e d t o t h e new a x e s 1 * T ,

zC

L e t LP b e p a r a l l e l t o t h e x - a x i s and m e e t TQZ i n L

a n d i n L * T hen LP » x a n d L^P “ x and

3B-K^« LL^ « h , o r X " X * h .

S i m i l a r l y y ■ y + k , a n d s - s ^ + /^ #

6 4 . R o t a t i o n o f a x e s . — L e t / /g * m g ,n g ;

/ ^ , m a , n g b e t h e d i r e c t i o n o o s i n e s o f t h e new a x e s X ^

Z ^ r e f e r r e d t o t h e o l d .

L e t P b e a n y p o i n t w h o se c o o r d i n a t e s w hen r e f e r r e d/ X /

t o t h e o l d s y s t e m a r e x , y , » a n d t o t h e new x , y , z .


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Draw XL p e r f a n & ie a la r t o t b o

plazLo X^OT anft U f p o r p o n d lo o la r

t o QX^* Thom on im x' , ML i s y

«HA PL I s s^ *

Sim oo t h e p r o J o s t l o a o f OP on OX I s oq?%al t o t h e

snm o f th o p r o j o o t lo m s o f 0H» ML, sad . PL o n OX wo hsw o

X " * /^jÿT * z

s a d s i m i l a r l y a . / ^/ / /

an d z * n ^ x + n g y + h@ z

w h io h a r o t h o r o q n lr o d f o r m n ls s f o r r o t a t i o n *

S lh o o f X g* m g .s g f a r o d i r o o t i o n

o o a in o o

« 1

^g^osi^»ong2 - 1

^^+m@8*ng3 - 1

and s ln o o t h o a x e s a r o m n t n a l ly p o r p o n d io u la r

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and s l n e o n^ «n 0 «B3 t h o d i r o o t i o n


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0 f OX, OX, aaA 0 2 «hem r e f a r r e t t o t h e a x e s

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w e h a v e -------------------------------------- — ■ - **3. . ^% 3 T ^ Z 2»3r" 3 ^

S g ? $ a rln g e a o h f r a e t l o n an d a d d iii^ a n t é c é d e n t s and

e o n s e a u e n t s y_________________________________, ___________:________ k '

3“ ^ * )* ♦ < 2 “ 3 ~ 3 - g r

f c a a n d ln ^ t h e d e n o m in a to r an d s u b t r a c t i n g a n d a d d in g

/ g ^ ^ S > m j 5* i^ * + a g * n g d a n d f a c t o r i n g w e h a v e

B u t e a c h o f t h e e x p r e s s i o n s i n p a r e n t h e s i s i s e g ,u a l

t o 1 i n th e f i r s t tw o t e r m s a n d t h e l a s t te r m i s e q u a l t o

s e r o an d h e n c e t h e w h o le e a q p r e s s io n i s e q u a l t o 1 and


91

and k» - 1, k « 1 1.

fh « T a lu e o f k d o o s n o t ohmn&e d o r l u g a o o r t a l n zo«

l o t i o n * T hat l a . I f k - 1 I t l a 1 f o r t h o r o t a t i o n , i f

on# ayatom l a r o t a t e d o o m p le t e ly i n t o t h e o t h e r “ 1 ,

> 3 • 0, ^3 • 0, • 0, Bjg « 1, *3 « 0, » i - 0, »2 « 0,n g « 1 an d from - (m g m g -a ^ g )k wo h a v e k « 1 .

I f t h e a x e a a r e r o t a t e d t h r o n g h 1 0 0 * , t h a t l a - a

e o l n e l d e w i t h ♦ * , t h e n * —1 , * 0 * ^ • 0 # * i ■ 0 »

# g • —1 , mg “ 0 , Û1 ■ 0 , a g ■ 0 , n g • —1 , a n d k « —1 #

S x e r o l s e a

1 . T ran aform t h e e q u a t io n x*+ 7y*e23B -® y-8 - 0 t o a a e t

o f a x e a p a r a l l e l t o t h e c o o r d i n a t e a x e a an d h a v in g th e p o i n t ( ! » — a a t h e o r i g i n .

8 . P r o v e t h a t t h e l l n e a 2 ^ p * 2 ^ « •— 2 ^ ^ 2 ^ fo r m a a e t o f r e c t a n g u la ra x e a . T r a n sfo r m t h e e q u a t io n xB + y2+ a2 - 1 0 0 s o a s t o r e f e r t o t h e s e l i n e s a s a x e a .

3 . T ra n sfo r m t h e e q u a t io n x?-7ak + $% y4.4yz-,3x+ 2y+ 16 « 0 t o t h e l i n e s g iv e n i n E x a a ^ le Z a s a x e a .

4 . Show t h a t t h e l l n e a

^ • J g ■ 5 ^ fo rm a s e t o f r e c t a n g u l a r a x e a . T r a n a fo im t h e a q u a t i o n x 0 +y2 +m2 « 8 1 t o t h i a s e t o f a x e a .


9 2

5# F la f t t h e é q u a t i o n o f t h e l i n e th r o u g h th e o r i g i n p e p p e n d lo u la r t o t h e l l n e a % * § “ x* ^ ^ an&t r a n s f o r m t h e e q u a t i o n « 0 t ot h e s e t o f r e e t a n g u la r a x e s fo rm ed h y t l ^ s e l i n e s .

O h l le u e C o o r d in a t e s

65* C o o r d in a t e s o f a p o in t* " - T he o o o r d ln a t e s o f a

p o i n t a r e t h e d l a t a n o e s f r e m t h e e o o r d l n a t e p l a n e s m easured

a l o n g t h e l i n e s p a r a l l e l t o t h e e o o r d l n a t e a x e s *

6 6 * P r o j e c t i o n o f a p o i n t o n a x e s * — I f a p la n e i s

p a s s e d th r o u g h t h e P o i n t P ( x , y » s ) p a r a l l e l t o t h e y s - p l a n e

an d m e e t in g t h e x - a x l s i n a p o i n t mho s e n - o o o r d ln a t e i s

%2 # v e may r e g a r d t h i s p o i n t a s t h e p r o j e c t i o n o f t h e

p o i n t P o n t h e x - ^ x l s s h e a t h e a x e s a r e o b l iq u e # I f

P g ( x g * y g # s g ) I s p r o j e e t e d u p o n t h e x - a x l s t h e x - e o o r d i

n a t e o f t h e p o i n t s h o r e t h e p la n e t o u c h e s t h e x - a x l s i s

x g and t h e p r o j e c t i o n o f P^^Pg on t h e x - a x l s i s x g -x ^ *

I f t h e p o i n t s a r e s i m i l a r l y p r o j e c t e d u p o n t h e y a n d a a x es

t h e n t h e p r o j e c t i o n s o f P%Pg o n y I s y g - y ^ and o n t h e

a x i s sg r ® !*

6 7 » G e n e r a l e q u a t io n o f t h e p la n e * — The p r o o f t h a t

Ax+%r+Gz*D « 0 i s t h e e q u a t io n o f a p la n e w as b a s e d on

t h e c o l l l n e a r i t y o f t h r e e p o i n t s # S in c e t h e c o n d i t i o n

f o r c o l l l n e a r i t y l a t h e sam e f o r o b l iq u e and r e c t a n g u l a r


93

o o o r d ln a t e s , t h e e q u a t io n • 0 i s t h e eqaar-

t i o n o f a p la n e f o r o h l ig n # o o o r d ln a t e s *

6 8 * P la n e th r o u g h t h r e e p o i n t s * — To f i n d t h e

e q u a t io n o f th e p la n e p a s s i n g th r o u g h t h o t h r e e p o i n t s

P g ( z 2 . y 2 , « 2 )« * p r o o « * 4

e x a o t l y a s we d id i n f an d t h e r e s u l t a s b e f o r e i s1

X 7 s 1

* 1 T l “ 1 1

% rz *z 1

*3 Tg *3 ^

and t h e I n t e r o e p t fo rm o f t h e e q ;a a tio n a s b e f o r e I s

a * Ç * f ”69* SqruatlOfts o f s t r a i g h t l i n e * — I n t h e e q u a t io n s

a—S5^ "n"

and n a r e p r o p o r t i o n a l t o

S c2 a. S 2 a . S :r ' r ' r

w h ere an d P j^ (x g ,7 g , s g ) a r e o n t h e l o o u s and

r i s t h e d i s t a n o e b e tw e e n a n d p g* T h at t h e s e e q u a t io n s

r e p r e s e n t a s t r a i g h t l i n e f o l l o w s im m e d ia te ly fro m t h e

p r o p e r t i e s o f s i m i l a r t r i a n g l e s *


94

P o la r C o o r d in a te s *

7 0 • C o o r d in a t e s o f a p o in t * — Tho l i n o OP draw n

fr o m t h o o r i g i n t o a n y p o i n t

P i # o a l l o d t h o xm din# v o o t o r o f ^

P* Tho p o s i t i o n o f t h o p o i n t P ^ ^

i # o o n p l o t o l y d o to r m in o d id io n wo

know t h o l e n g t h o f t h o r a d iu # v o o t o r

t h o a n g l e ZOP, a n d t h o a n g l e h o tw o o n

t h o p ia n o # ZOZ a n d POZ* T ho#o o o o r d in a t o s a r o o a l l o d

t h e p o l a r o o o r d i n a t o s o f P and a r e d e n o te d h y

Wo may e x p r e s s t h o r o o t a n g q l a r o o o r d in a t o s o f p i n

te r a m o f i t s p o l a r o o o r d in a t o s *

Draw PQ p o r p e n d io u la r t o t h e x M P la n o a n d QR p o r p o n

d i o n l a r t o €EZ*

T h en X • OR » OQ od # ^* .f ay

0 ^ « OP s i n ^ » s i n # and h e n e e % « s i n ^ o o s j*f

y ■ QR s i n s i n ^ s i n f•

s * PQ ■ c o s

Wo may a l s o e x p r e s s t h o p o l a r o o o r d in a t o s o f a p o in

i n t o r n # o f th o r o o t a n g n la r o o o r d in a t o s o f t h a t p o in t *

S in o o ^ i s t h o d i s t a n o e fr o m t h e o r i g i n t o P (x $ y » a ) t h e s

a r o


95

C « V3t?»3r*+»*t ® "

&c«rol8«8

1 « ? ln d th « r e f t a n g u l e r « o o r d la a t e a o f & p o i n t i f i t s p o la r o o o r d in a to o a r o i

( a ) ( 2 ,3 0 * 4 6 « ) ^ ,(D ) ( 3 ,1 2 0 0 , 6 0 0 )( a ) ( 4 , 3 0 0 , 1 3 8 * ) .

S* TSmlns a n o b l lq u o a y a to m o f o o o r d ln a t o a p r o v e t h a t t h o l i n o a o f I n t e r a o o t l o n o f t h r o e p la n o a " A%oByoCaoD»0 ,

• 0 * Ag%+Bgy+Gga+% » 0 a r o p a r a l l e l i fA B CAi Bi Cl

®2 ®2

0.3 * U a in g a a y sto m o f o b l i g a o o o o r d in a t o s p r o v e t h a t th o tw o p la n o a iz+ B y*C aoP « 0 , A i% -^Biy+C iaoD i « 0 a r o p a r a l l e l

i f Ix - Ixc

4 . F in d th o d i s t a n o e b e tw e e n t h o tw o p o i n t s w h o a o p o la r o o o r d in a t o s a r o r e s p e o t i v e l y ( 3 ^3 0 0 4 5 0 ) , and (2 )1 3 5 0 ,2 7 0 ^ )#

5* B o s o r ib o th o p o s i t i o n o f a l l p o i n t s i n a p a o o f o r w h io h

X • 3♦ - 120®X • 3 . y - 3r • 3 ) 45 • 30® .

6 « V hat a r e t h o d i r o o t i o n o o s i n e s o f th o r a d in s v e o t r o w h io h l i e s i n th o x y ^ l a n o and w h io h m akes an a n g le o f 5 0 0 w i t h t h o p o s i t i v e y ^ -a x is*

7* F in d th o d i r e c t i o n o o s i n e s o f t h o a n g le s u b te n d e d a t th o o r i g i n b y t h o l i n o se g m e n t J o in in g t h o tw o p o i n t s ( ^ . - e . - 6 ) , ( - 1 4 , 6 , 2 ) .


96

8 * Show t h a t h y p r o p e r l y o h o o a lu ^ o h llq :iio a x e s t h e e g o a t l o n o f a a e l l i p s o i d may h e p u t i n t h e fo rm %*+y^ + # 8 m a . 8 # S h ow t h a t a a l a f l n i i ï é num ber o f s y s t e m s o f s u i t a h l e a x e s earn h e fou n d *

9 * D e te r m in e w h e th e r t h e e q tu a t io a o f a h y p e r h o lo id o f o n e s h e e t e a n a lw a y s h e p u t i n t h e fo r m x 8 +y8 —s 8 • a ^ 8 h y p r o p e r l y o h o o s in g o h l i d u e a x e s *

1 0 * B y l e t t i n g t h e t h r e e e o o r d l n a t e p l a n e s o o ln o id e w i t h t h r e e f& o e s o f a t e t r a h e d r o n p r o v e t h a t l i n e s e o n n e e t i n g t h e m id d le p o i n t s o f o p p o s i t e e d g e s m eet i n a p o in t *


97

C h a p te r 7 1

Tam gentg an d B o rm a la t o t h o C o n lo o id

7 1 . S lm p lo o t fo r m o f e q u a t io n o f o o n l o o l d , — The

o q ;a a tlo a

d3c?^Jar*+Ca*>Cw*+k * 0

r o p r o s e n t s a l l o l a a s o s o f o o n l o o i d a . I f w ■ k • 0 wo

h a v o a n o q n a t lo n o f t h o fo r m 2 ^ 2 ^ • 0 , I f w », • df t» 0»

0 we h a v e t h o fo r m 35* * e % » l , i f C * k # 0 w eÏÇ ^ - 0»

h a v e t h e fo r m £ * » 4 p g , a n d i f 6 ■ w « 0 w e h a r ea » - h »

th o fo r m Ax*+ky®+k • 0 #

72. D o f i n l t l o n o f t a n g e n t p ia n o t o a o n r f a o o .—

Two t a n g e n t l i n e s draw n t o a s u r f a o o a t a n y p o i n t on i t

d e te r m in e a p ia n o s a i l e d t h o t a n g e n t p l a n e , t h e p o i n t

b e in g o a l l o d t h e p o i n t o f t a n g e n e y o r t h e p o i n t o f o o n

t a o t .

73. S g o a t io n o f t a n g e n t p l a n e , — L e t ? i(3 c ^ » 7 i» a ^ )

h o a n y p o i n t o n t h o s u r f a o o

lx^ehyt>G z^^2w s4>k « 0 ( 1 )

L e t n s p a s s tw o p la n e s y ■ y x t h r o n g h a n d p a r

a l l e l t o t h e s3 D -p la n e . and x * a n d p a r a l l e l t o t h e

y z - p l a n o .


98

T b s p l a n e s y • y x an d % * z% e u t fro m t h e o o n lo o ld

tw o o o n lo s o o t I o n s v h l o h h a r o f o r t h e i r e q u a t io n s

d z .^ * h y ^ + 0 s^ + 8 m s+ k « 0 ,^ X- Xx ■ 0

a n d Ax^*By^ 3^ C s^ *8w s*k • 0^ ( 3 )

MTX " 0

a n d t h e l i n e s t a n g e n t t o t h e s e o u r v e s a t t h e p o i n t ? x

a r e

B y x y + 8 z x a * # ( st^ sx ) * ^ 1 ^ * ^ " 0X-4CX • 0

a n d dZxZ4>CzxS’Hr(B4>sx)'*‘3 y^ 3 «h « 0

j - n ■ 0

U )

( 5 )

L e t t h e d l r e o t l o n o o s i n e s o f t h e l i n e r e p r e s e n t e d

h y e q u a t i o n s ( 4 ) h e "x* ^ and t h e d l r e o t l o n o o s i n e s

o f t h e l i n e r e p r e s e n t e d h y e q u a t io n s ( 5 ) h e g* m^, n g .

T h en h y

Czx+* 0 0 Byx

0 X ' 1 0

Ù O Z i+ # G zi*w A%x 0•»4 1 o ' 0 0 ' 0 1

H«B«« I t f o l l o w s I m a s d l s t o l j t h a t x * ^ » n d mg « 0


99

an d h e a o * we h a v e t h e f e l l o w in g p r o p o r t i o n s

and

^ g : m g : n g « -(Csj^-Mr) : 0 : l a ^

and t h e e q u a t io n o f t h e p la n e i s

* -* ! 3TT1 « 10 Csi+w “ 1 • 0

- C s y # 0 J

fro m w h lo h w e h a v e

( 3 ^ 1 ) ( » -S iX C s i* w )^ • 0

o r ( Ix^aî-^âJCj^* ) ( C s3 4W ) ♦ ( XOz j ^w) ♦ ( s -s j^ ) ( Csg^+w) 2 «o

a n d h e n e e

AXiX*B3r2.3r*Cz2,8-»w-(âx 8*Byx8*Cz3_8>z^w ) • 0.

S u b s t i t u t i n g Sw s^^h f o r —

we h a v e

z+s^) wk ■ 0

w h ic h i s t h e e q u a t io n o f t h e t a n g e n t p la n e t o ( 1 ) a t

t h e p o in t

74$ S p e o i a l fo r m o f e q u a t io n o f t a n g e n t p la n e t o

e l l i p s o i d and t h e h y p e r b o lo id s # - . L e t u s c o n s i d e r t h e


100

• I l i p t 0 l 4

ÿ 0 ÿ " 1t h # p l M f « t w h lo a 1#

(1)

^ . f i x . “ 1

»% th# p«lnt (*x»3^**l^*

# r i # l à g t â e ê q a à ü lô a o f t h # t ë h g o a t p la à o l à t h e

f e r m Vm+myeam ^ p w e W v e ^ : a : h * The

l e n g t h o f t h e p e r p e n A lo a la r 2, f r o * t h e o r i g i n u p o n

p la n e ( 2 ) l a g i v e n h y t h e e q u a t I o n

»»

*^ 8

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S l B l l w l y " b * i^ • anC o ^ * “ »

H.na« - a ÿ ♦


iOl

B u t fr o m (X ) o lm o # o n t h # o l X l p o o l i

a n â h o n o e * # * *

B n t p " .^ * n y ^ n # an d t h e r e f o r e

/x+aqr-wÉi* • ± y » Ç * ^ P Ç K e ^ ?. J ‘ ,1 ' '' ■ > :? ■■ _

r e p r e e e n t e t h e e q u a t io n o f t h e t a n g e n t p la n e t e t h e

e l X l p e e l d f o r a l l v a l o e e o f m« m*

I f t h e e q u a t io n o f a o o n i e o l d 1#

t h e n t h e l e f t member o f t h e e % n a tlo n a n a lo g o u e t o ( 3 )

l e

an d h e n e e t h e e q .m t io n o f t h e t a n g e n t p la n e o f t h i s

o o n i e o l d l e

/^ ♦ n r - M w • i v 5 Ç ^ 5 A 5 ± 5 n 8 e

79* D l r e o t o r e |h e r e o f e l l l p e e l d an d t h e h y p e r b o lo id .

We m l e h t o f i n d t h e l e o n e o f t h e p o i n t eommon t o t h r e e

e n t m a l ly p e r p e n d le n la r p la n e # e a o h t a n g e n t t o t h e e l l l p -

e o ld #


108

tJ«l4ae t h # Q f 7 4 l # t # f t h #

p l » « a b #

^ixtPajr*»!» « ( i)

^g3:4#gy4ag8 • (2 )' . 0 ••* - » . J

( 8 )

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p tV P « B 4 i« iU .a v w e îM ^ r e g a r d ^ , X g # ^ a s t h e A l r e e t i e a

e o s i a e s o f t h e z - # z l # $ h e a r e f e r r e d t o t h e n o r m a ls on

t h e s e p l a n e s a s a x e s * S i m i l a r l y # ] , , mg^ i ^ may h e r e

g a r d e d a s t h e d i r e s t I o n e o s l n e s o f t h e y ^ -a x is» and r i# n g *

mg a s t h e d i r e s t l o # e o s l a o f o f t h e s - a x l s a& en r e f e r r e d

t o t h e n o r m a ls on t h e s e p l a n e s a s a x e s .

H en oe * 1

m^^4e%^emg * 1

nj^^+ng^+ng^ « 1 .

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( 2 ) a n d ( 3 ) and a d d in g

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o r Î + z * ( n^ 2 +ng?^ ng 2 )


108

♦* ( 4 )

«nd t h e r e f o r e

*2n ( “ #f*&**»*.

But slùee th e plahes e re m tu ftll^ yer»emdleulmr

A i ^ * 4 ^ * 4 " s ■ ®

A »1*A >^*'4*3 * °

an d we g e t # e » f l u e l r e s u l t

x^ +yS+aB « #B +bB +e2

a a d t h e l e e o s l e m s p h e r e w h ie h i s s a i l e d t h e d l r e e t o r

s p h e r e o f t h e e l l i p s o i d *

I n t h e e a s e e f t h e h y p e r t i e l e i d s t h e e d u a t l o a s o f th e

t a n g e n t p l a n e s h tT t t h # f e r n

./^engr+m h' - /e % ^ ± b ^ n A b s ^ ^

a n d heneeaAeyë+m* » a* n h* ^ s^ ^

i s t h e e q u a t io n o f t h e d l r e o t o r s p h e r e o f t h e h y p e r h e l—

o ld s *


104

T6 * f e r m @f t h # •q .u a t lo n o f tm a g e n t p la n #. > - ■ ■ . - . -

t o p arab oX olC «-« L o t t h o o ^ m t l o n o f th o p a r a t o l o t d t o

« 0 (1)

a n d %hm o ^ m t lo m o f t h o t a n g o a t p ia n o a t (x i#y i*a% )k ! •

- O (8)

B u t o q u a t lo a ( 8 ) l a o q n lv a l o n t t o a n o q n a t lo n o f

t h o fo r m

y&["#ogr*mm - p (a)

w honoo

f h o l e n g t h o f t h e p o r p o n d io u la r p fr o m t h o o r i g i n

t o th o p ia n o ( 2 ) l a g i v e n b y t h o o q n a t lo n

a n d /& »

R onoo wo h a v e


1 0 6

Bo* 1 1 # » on (X ) a a 4 h e n # #

^ * ÿ - zap «la

#nA h#m## Xx^ngrenn ■ * |2 . ) ( 4 )

1 # a t a n g e n t p la n e t o t h e p a r a b o lo id f o r a l l v a ln e # o f

JL^ ■ , n*

I f t h e e q n & tlo n o f t h e p a r a b o lo id i s i n t h e fo r m

S-S ■ *'A - - 5 j i g (C ) an d B - ( 6 ) .

S X im in a t in g A and B fr o m (4), (5) a n d (6 ) w# o b t a in

B& ^m nFââ ■ —p ( a % ^ + n fb ^ ) .

7 7 * L o e u s o f i s t e r s e o t i o n o f t h r e e m n t n a l ly p e r p e n -

d i e n l a r t a n g e n t p la n e # t a t h e p a r a b o lo id * — l e t u a o o n s id e r

t h e t h r e e m a t n a l ly p e r p e n d ie o la r t a n g e n t p la n e #

îm ^x*m in^y+n% 2m * -p(a*j£2-hb*m j^2) ( 1 )

- -p(a^£2^b2«^2 ) ( 2)

/^n^x^msn^â^^s * -p (a ^ + b ^ :y 2 ) (3)

Wa may r e g a r d X%, a s t h e d i r e o t i o n e o # i n e # o f

t h e x - m x i s , mj^, m g, mg a # t h e d i r e o t i o n e o s i n e # o f t h e y -


106

M è o ^ é» . w th . *-

û d . whBB wttiimi % , t h . n o x m I b t o t h . tawa^gont p l « e i . .

T h t s s lA M t h M . t o n g o n t plm noie a r . m atm alljr p .V 9 « ^

^ m lm r

«a* !• &i^+ng»om@ • 1*

A W ln a . t r r .% » o m ^ a # m # # t . w # ^ t l ( w

(@) w . h a v .

( )y * -*(a$+h^)

a a t .o |^ (ik ^ o h ^ ) ■ O

1 . a p l a n , w h lo h . l a t h . r .g ? i l r .& loe^ as«

X t w . now . . m a l & . f t h . h y p e r b o l io p a n a h o lo ld

-lâî- L ' * tpa

w . fliWk t h . l . o W 1 . t h . p l a h .

m.p(a^-h*J • 0*

7 8 « # 0 9 * t l o a o f H a . n o r m a l t e t h . e l l l p a e l d and t— -.., y ■■ - „-r- '--' -'■• -

t h . h j p o ' h h l . l d . * - * A l i n . p o r p a n d lo tO a r t o t h . t a r e n t


107

p lA a « t h f p o i f t t o f 1 » o a U o d t h # & @ « 1. &F '

1 1 m# t o t h # o m rfo o o *

l o t u s o o u s ld o r t h e o o n i e o l d

t% B*PyB+O s^*0#s+D » 0 ( 1 )

an d t h e p la n e t a o ^ e n t t o I t a t I s

" 0 ( 8 )

T he e q u a t io n o f t h e l i n e no^rnal ^ t h e o o n i o e l d a t

t h e p o i n t % i # y i , s i I s

» -g c i y ^ i n - # i

T h u s , t h e l i n e n o z n a l t o t h e e l l i p s o i d o r h y p e r b o lo id

i s

« 1 r -T i e-g la * y i ‘ *13 b?

F o r a n y p a r t i e u l a r v a lu e O f ea o h f r a c t i o n v e o b t a i n

a d e f i n i t e s e t o f v a l u e s o f L e t t ie p u t e a o h

f r a e t i o m e q u a l t o k#

« - 2 . » .

B 5^ ?

w ^ + n ! ♦ ^ * j j V g . * + j » £ * i î . 1a » b « oZ (a * * k ) ^ (b 8 ^ k ) 2 (© « ♦ k )*


1 0 8

V« # # 4 « • b o t h member# o f t h i #

• q m t l o m b y « * ♦ !? )* w e g e t e a e g m a t ie a o f

t h e é iâ c th ftCigrèw. t a le é à C t h e r t ir b r e e a e k f ^ e t l o m hme

s i x y a ltiem we h a v e # l x l l a e a fr e m a n e x t e r i o r p o i n t

n o r m a l t o a n e l l i p s o i d o r l iy p e r b e lo id «. '"t- ■'

6 0 * #m W hea o f # e r m # l# f f e » e x t e r i o r p o i n t t o t h e

p a r a b e l e l d . — T he p la n e t a n g e n t t e t h e p a r a b o lo id

> n ÿ a ÿ - 4 , . . _ (1)

a t p o i n t t % i # y i* e i ) l e

n ZlZ m 2p(s>s^)ta# *# 2 b 8- ■ - - -

a n d h e n e e t h e e q u a t i o n o f t h e l i n e n o rm a l t o t h iw p la n e

a n d p a s s i n g t h r o u g h som e e x t e r i o r p o i n t (%2 * y g » e g ) l e

»gr«i y r n » r ^ i z : s r

F o r a n y p a r t i e u l a r v a lu e o f e a o h o f t h e s e f r a o t l o n s

w e ^ e t a d e f i n i t e s e t o f v a l u e s o f x ^ ^ y i^ s ^ * e » e h,r ii.

f r a o t i o n e q u a l k«

T h en F a r y g _X I y i - 8 pmZ bZ


109

a a d

Jt * sB a t « la * # : % . y i . % l i « " « ( D - ■t»«i - «

*» S»

2

(#*4*)» (b*4k)*I ■- -: c a.-% ::■: , j.1: r. J J _ . • "

b a tik ja a m b a r# j f t h l # a ^ m tlo iB 1 ^

i a d |^ ) ^ ( ^ d 4 .3( ) d ^ g e t a a e ^ m t l o a o f t h a f i f t h d e g r e e In

k an d . ^ e ^ e f f r e v e h a r e f i r e l i n e s f r o # a n e x t e r i o r p o in t

n o x a a l t e t n * p a r a b o lo id .

@1* L in e s t to z v a l t o e o n e a and o y l l n d e r a . — T he e x e r

t i o n o f t h e t a n g e n t p la n s t o t h s e o n s

ÿ i # ± $ - «

s i n e s t h s t a n s o n t p la n e p a s s e s th r o u g h t h e o r i g i n'if '.

and t h e p o i n t t h s l i n e d e t e r m in e d h y t h e s e t e e

p o i n t s l i e s I n t h e t a n g e n t p l a n e a n d m o r e o v e r l i e s I n: - TS -

t h e e o n s . H enoe a t a n g e n t p la n e t o a e o n s e o n t a l n s a


110

r - .jv. - -# # # % $ êm t# % %hm: & « m i l l i a # # $ # t h e e o æ 1 #5 #

fJ* * 9 r v ■» B • 0 (?);->?J . *" •<•' m 'V #1 .- k-:

» -* 1) “ o (4)

h t ^ t h è » 9 # l a t X x » 7 x * * i*

E%# 't# : t h # «m rfm ee r e p r e s e n t e d b y e g n e t lo m' ■ _ ■ . tV rjV - ,-.■■■■'

( 4 ) i i e l e m r l y

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astêrelses

S$228-A

1 . eq 9 # t i# m e f t h e t a n g e n t p la n e t o t h e e l l l p —, 0 1 4 t h e p o i n t ( 2 , 1 , - 8 ) .

'1 v'# . V la 4 t h o e ^ im t la n e f t h e n o r m a l l i n e t o t h e e n r f a e e^ * e * # - 1 •* ( ? , - i . 2)f , ?& t .. --

t h è O q in o tl^ ^ ^ t h e t a n g e n t p la n e to the a u r f a o e e JO # t t M o r i g i n #


I l l

4 * ? i a â $ h # 1 1 a # a o v n a l %# t h # # n r f» # #

5# WisA t h # e^ a a tlm m # # f t h * # 1 # » # # t # t h eX % w # h # w r w r m lW t # tk * -

p la n e 6% f3gr*e-4 • 0 *. ’ . ^ ■ -i ‘r-/ ' ..*f . •'- '• •■î-T' • -■ : TV-4 # fW K t h e m a a ^ e # W k t h e . U s e mowmtX t e t h e #%*$%## 8 %B-0 y% 1 @1 È a t t h e p o i n t ( 4 , 3 , 1 ) .

“ “ ¥ < 1 ' ¥ • “ ?A* Y in t i t h e e v m t le m * ? h ^ t h # t n n g e n # ^m m ee^ t e , t h # hpw p # F h e i# 1 4 l^ ^ ) L A # t * + ^ h - 4 4 P ft a t t h e p o ^ t e e h f * e x - S ,y " - 1 #

9 . F l h t t h é 0 (sx» t l e a n o f t h e t g j A è à t j ü ü i& 4 a /^ t h é o o n l o ^ d t x f t - 4 y 2 * 5 8 8 * 6 à e l • 0 a t t h o p o i n t s e h e n o x y ■ 5 .

IÔ* y in f t t h e o f t o a t l é a t o ;f # h é t a n f t f n t p lh n e # t e t h # e l l i p s o i d ' w h ie h are" p a r a l l e l t o t h s p la n e a x 4h y * o s + d - 0 .

1 1 $, yiz&d t h e l e o n e e f t h e i n f o n a e e t l e n p f ^ r o e v a t n a H y p e r p e n d le n la r p l a n e e w h i # L a n # t a n g ^ t t o ^ 8 s .

12# P r o v e t h a t t h e p lfu ie x e o s ^ e j e o s ^ e a e e s ^ p^I s t a n g e n t t o t h # p i i n a t e l o l d â t ^ * 4pm i f p^ > a h e o s h ^ e h # e s e # ( ^ ^ * ^

ftrom p B

1 . F in d t h e e q n a t l e n o f t h e t a n g e n t p la n e t o t h e s n r f a o e x 8 ej^ ft" 2 s 8 * 0 a t ( 4 1 , 3 ) •

2 « F in d t h e e q n a t io n o f t h e n o r m a l l i n e t o t h e s p h e r e xB + y3+ a8 « 9 a t t h e p o i n t ( 2 , - 3 , 1 ) .


112

s» 71b4 th* fQiÉpLtloa of th# tangent plan# te th# sur— f#*# 0 at th* point (4,&,-9).

4* y&#â th* *qputtlons of th# line norml to th# sur-fa** • 1 0 at th# point

9* yjgt th#*<Tiatlon» of th# tangent planes to the ooa-1**14 %'—C' * Ëg » 1 whloh are parallel to the planea%*hyf*##4 ». 0*

6 * FinA th# equation# of the line normal to the eurfao# hadWyO IS* at th# point (4,1,6) •

7$ Flnâ th# equations of th# tangent planes to the # 1 1 Ip— sold ^ • 1 at the points where x • 3, 7 » 6 #

8 « Find th* equations of the tangent planes to the hyper- h o l o i d . 44 » 0 at the points where x * 3, y •

9. A sphere Is tangent to the plane x- Sy z « 7 at the point (8 ,—lv-1 ) and passes through the point ( l , l , - 0 ).Find its equation.

10* Find th# equations of the tangent plane and normal lln*<t# th# ellipsoid 3x8+5y8+2z8 * 41 at the point (1,8,-»)*

IIV Find the eoerdinates of points on the ellipsoid 4 k»o4 j 8 o1 8 s» * 8 8 the tangent planes at whieh points make #t^al Intereepts on the ooordlnate axes.

18, , Pettrmlm# th* loeus of a point which moves so as al#ay# t* he eqaally distant from two given straight lines.

group C


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6« The a e r w l e t im y p o ^ F e f e a e X Ü p e e ld m e e t# a p r l a e i p e l p l e a e l a G . S h e # t h e t t h e l o e û s e f t h e m id d le p e l â t e f FG 1# e n e l l l p e e l d #

T* S h e # t h e t e e e e t i e a e f e h j p e a b e l e l d mede h y e p l e a e p e e e l l e l t e e a e le m e n t o f t h e e e y m p t e t le e e a e l e e p e r e h e l e .

S . S h e # t h e t t h e g e n e r e l e e a e t l e a o f e e e a e r e f e r r e d t e t h r e e e f l i s g e a e r e t o r e e e e e e r d l n e t e e e z e e l e P f e t S e x♦Baqr ■ A*

e # Frem t h e e q a e t l o a e e f t h e t e a g e e t p l e a e e t e t h e e e a e e a d t h e h ÿ p e r h e l l e p e r e h e le lA » p r o e e t h e t e h e a e p o i n t F l e m ede t e move e l e a g % g e a e r e t l a g U n e o f e o e a e t h e t e a g e n t p l e a e e t F rem elm e e t e t i o a e r y b u t e h e a e p o i n t F l e mede t e m ore e l e a g e g e n e r a t i n g l i n e o f e h y p e r— b e l l e p e r e b e l e l d t h é t a n g e n t p la n e a t F r e v o lv e e a b o u t t h e g e n e r a t i n g l i n e #


115

1 0 * t h # 9 I 0 9 0 z tm y + m # # p w i n b e t e o g e n tt e t h e e l l i p e e i f t

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11^ P w e w e ^ h e * o t X l j f é e t e e l d e w h ie h p e e # t h r e w g h e e v e n given p o in te pee# threw gh ehethew fliseâ po in t*IP . Phew th a t th e th re e ■ h tne |3y p erp en d len ln r tan g en t p lane# to th e e l l lp e e ld 3 ^ e M # % isè é ÿ e eè 'r ehth e emhere e ,a0eh*ee»7


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p o in t w i t h r o a p a e t t o t h # o ^ n ia a id ( l )

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a n d a ln o # ( 2 ) p a s s # # th r o u g h % i* y i# a i

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p l s n e w i t h r e e p e e t t% t h e o o n e w M o h d o e s n o t p e e # t h r e o g t

t h e o n i g l # #r Mf ■ ’ ' •« . •

I * w • p t h e 0 o o n 6 h # t e s e f t h # p e l # e r e i h d e t e r *

m i n e i e e n d h e n e e th e f p i e n e # t h r O e g h t h e w e w te x e f e e o n -

i o e l s n r f a o e h a s n o p o l e w i t h r o s p e o t t o t h e t s o r f a o e »


120

E z e r o l# # #

1 * liJaA t h e e c p iA t l^ e f t h e lo X i^ p le& e e f t h e j^ e ia t U'

' J #r 4 -# i. jj. ■ '*‘ ''• - : ’ '

9« # h e # t h e t t h e I tm e OOP j e lm la * t h e e e a t e f 0 e f m s p h è r e t e m p o i n t ? l e p e r p e a d lo n le r t e t h e p e l e r p l e a e e f P«

4 « Show t h e t t h e d l e t e a e e e ^ f t e e p e i n t e < n e# th # e e a t e r e f # s p h e r e e r e p r e p e r t i e a e l t e t h e â i e t e a e e e e f e * e h fr o m t h e p o la r p l e a e e f t h # e th e r *

9* B reT o t h a t eagr tw e p l e a e s e o t i o a e e f a o e a i o o i e and t h e p o l o s e f t h o s e p la n e s l i e e n a n o th e r o o n ie o ld *

6 * Show t h a t t h e d l s t a n e e # o f hoe p o i n t s from t h e e o n t e r o f a s p h e r e a r e p r o p o r t io n a l t o t h é è i s t a n o o s o f e a o h fro m t h e p o la r p la n e o f t h e o th e r *

7* Shew t h a t t h e l o o n s o f t h e p o in t o f i n t e r s o o t io n o f t h r e e m a t n a l ly p e r p e a d ie u la r ta n g e n t p la n e s t o a p a r a b o lo i d i s a p la n e *

S* ProTO t h a t t h e lo o n s o f t h e p o l e s o f a s e r i e s o f p a r a l l e l p la n e s i s a s t r a i g h t l i n e th r o n g h t h e o e n t e r o f t h e o o n ie o ld *

9 * A l i n e th r o n g h t h e p o in t P , m o o ts th o o o n ie o lAi n t h e tw e p o i n t s p # an d p g* z n o th o r l i n e th r o n g h? ! m e e ts th e o e n i e o i d i n B g an d P5 * The p la n e s t a n g e n t t o t h e o e n i e o i d a t Pg and P g d e te r m in e a l i n e

a n d t h e p la n e s t a n g e n t t o i t a t P g and P g d e t e r -m in e a l im e > ^ * P r o v e t h a t a n d X fr im t e r s e o t o ra r e p a r a l l e l .


U l .

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124

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