Circle - Best Approach

Circle(Sheet)

Best Approach

Manoj Chauhan Sir (IIT Delhi)

Exp. More than 13 Years in Top Most Coachings of Kota

No. 1 Faculty of Unacademy,

By Mathematics Wizard

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Maths IIT-JEE ‘Best Approach’ (MC SIR) Circle

KEY CONCEPTS (CIRCLE)STANDARD RESULTS :

1. EQUATION OF A CIRCLE IN VARIOUS FORM:(a) The circle with centre (h, k) & radius ‘r’ has the equation;

(x h)2 + (y k)2 = r2.(b) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre as:

(g, f) & radius = g f c2 2 .

Remember that every second degree equation in x & y in which coefficient ofx2 = coefficient of y2 & there is no xy term always represents a circle.If g2 + f 2 c > 0 real circle.

g2 + f 2 c = 0 point circle.g2 + f 2 c < 0 imaginary circle.

Note that the general equation of a circle contains three arbitrary constants, g, f & c which correspondsto the fact that a unique circle passes through three non collinear points.(c) The equation of circle with (x1 , y1) & (x2 , y2) as its diameter is :

(x x1) (x x2) + (y y1) (y y2) = 0.Note that this will be the circle of least radius passing through (x1 , y1) & (x2 , y2).

2. INTERCEPTS MADE BY A CIRCLE ON THE AXES :The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinate axes are

2 g c2 & 2 f c2 respectively..

NOTE :

If g2 c > 0 circle cuts the x axis at two distinct points.

If g2 = c circle touches the x-axis.

If g2 < c circle lies completely above or below the x-axis.

3. POSITION OF A POINT w.r.t. A CIRCLE :The point (x1 , y1) is inside, on or outside the circle x2 + y2 + 2gx + 2fy + c = 0.according as x1

2 + y12 + 2gx1 + 2fy1 + c 0 .

Note : The greatest & the least distance of a point A from a circlewith centre C & radius r is AC + r & AC r respectively.

4. LINE & A CIRCLE :Let L = 0 be a line & S = 0 be a circle. If r is the radius of the circle & p is the length of theperpendicular from the centre on the line, then :

(i) p > r the line does not meet the circle i. e. passes out side the circle.

(ii) p = r the line touches the circle.

(iii) p < r the line is a secant of the circle.

(iv) p = 0 the line is a diameter of the circle.

5. PARAMETRIC EQUATIONS OF A CIRCLE :The parametric equations of (x h)2 + (y k)2 = r2 are :x = h + r cos ; y = k + r sin ; < where (h, k) is the centre,r is the radius & is a parameter.Note that equation of a straight line joining two point & on the circle x2 + y2 = a2 is

x cos

2 + y sin

2 = a cos

2.



6. TANGENT & NORMAL :(a) The equation of the tangent to the circle x2 + y2 = a2 at its point (x1 , y1) is,

x x1 + y y1 = a2. Hence equation of a tangent at (a cos , a sin ) is ;x cos + y sin = a. The point of intersection of the tangents at the points P() and Q() is

2

2

cos

cosa

, a sin

cos

2

2

.

(b) The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at its point (x1 , y1) is

xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.

(c) y = mx + c is always a tangent to the circle x2 + y2 = a2 if c2 = a2 (1 + m2) and the point of contact

is

a m

c

a

c

2 2

, .

(d) If a line is normal / orthogonal to a circle then it must pass through the centre of the circle. Usingthis fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1 , y1) is

y y1 = y f

x g1

1

(x x1).

7. A FAMILY OF CIRCLES :(a) The equation of the family of circles passing through the points of intersection of two circles

S1 = 0 & S2 = 0 is : S1 + K S2 = 0 (K 1).

(b) The equation of the family of circles passing through the point of intersection of a circleS = 0 & a line L = 0 is given by S + KL = 0.

(c) The equation of a family of circles passing through two given points (x1 , y1) & (x2 , y2) can be writtenin the form :

(x x1) (x x2) + (y y1) (y

y2) + K

x y

x y

x y

1

1

11 1

2 2

= 0 where K is a parameter..

(d) The equation of a family of circles touching a fixed line y y1 = m (x x1) at the fixed point (x1 , y1) is(x x1)

2 + (y y1)2 + K [y y1 m (x x1)] = 0 , where K is a parameter.

In case the line through (x1 , y1) is parallel to y - axis the equation of the family of circles touching itat (x1 , y1) becomes (x x1)

2 + (y y1)2 + K (x x1) = 0.

Also if line is parallel to x - axis the equation of the family of circles touching it at(x1

, y1) becomes (x x1)2 + (y y1)

2 + K (y y1) = 0.

(e) Equation of circle circumscribing a triangle whose sides are given by L1 = 0 ; L2 = 0 &L3 = 0 is given by ; L1L2 + L2L3 + L3L1 = 0 provided co-efficient of xy = 0 & co-efficient ofx2 = co-efficient of y2.

(f) Equation of circle circumscribing a quadrilateral whose side in order are represented by the linesL1 = 0, L2 = 0, L3 = 0 & L4 = 0 is L1L3 + L2L4 = 0 provided co-efficient ofx2 = co-efficient of y2 and co-efficient of xy = 0.

8. LENGTH OF A TANGENT AND POWER OF A POINT :The length of a tangent from an external point (x1 , y1) to the circle

S x2 + y2 + 2gx + 2fy + c = 0 is given by L = x y gx f y c12

12

1 12 2 = S1.

Square of length of the tangent from the point P is also called THE POWER OF POINT w.r.t. a circle.Power of a point remains constant w.r.t. a circle.Note that : power of a point P is positive, negative or zero according as the point ‘P’ is outside, insideor on the circle respectively.



9. DIRECTOR CIRCLE:The locus of the point of intersection of two perpendicular tangents is called the DIRECTOR CIRCLE of the

given circle. The director circle of a circle is the concentric circle having radius equal to 2 times theoriginal circle.

10. EQUATION OF THE CHORD WITH A GIVEN MIDDLE POINT :The equation of the chord of the circle S x2 + y2 + 2gx + 2fy + c = 0 in terms of its mid point

M (x1, y1) is y y1 = x g

y f1

1

(x x1). This on simplication can be put in the form

xx1 + yy1 + g (x + x1) + f (y + y1) + c = x12 + y1

2 + 2gx1 + 2fy1 + cwhich is designated by T = S1.Note that : the shortest chord of a circle passing through a point ‘M’ inside the circle,

is one chord whose middle point is M.

11. CHORD OF CONTACT :If two tangents PT1 & PT2 are drawn from the point P (x1, y1) to the circleS x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact T1T2 is :xx1 + yy1 + g (x + x1) + f (y + y1) + c = 0.

REMEMBER :(a) Chord of contact exists only if the point ‘P’ is not inside .

(b) Length of chord of contact T1 T2 = 22 LR

RL2

.

(c) Area of the triangle formed by the pair of the tangents & its chord of contact = 22

3

LR

LR

Where R is the radius of the circle & L is the length of the tangent from (x1, y1) on S = 0.

(d) Angle between the pair of tangents from (x1, y1) = tan1

22 RL

LR2

where R = radius ; L = length of tangent.(e) Equation of the circle circumscribing the triangle PT1 T2 is :

(x x1) (x + g) + (y y1) (y + f) = 0.(f) The joint equation of a pair of tangents drawn from the point A (x1 , y1) to the circle

x2 + y2 + 2gx + 2fy + c = 0 is : SS1 = T2.Where S x2 + y2 + 2gx + 2fy + c ; S1 x1

2 + y1

2 + 2gx1 + 2fy1 + c

T xx1 + yy1 + g(x + x1) + f(y + y1) + c.

12. POLE & POLAR :(i) If through a point P in the plane of the circle , there be drawn any straight line to meet the circle

in Q and R, the locus of the point of intersection of the tangents at Q & R is called the POLAR

OF THE POINT P ; also P is called the POLE OF THE POLAR.(ii) The equation to the polar of a point P (x1 , y1) w.r.t. the circle x2 + y2 = a2 is given by

xx1 + yy1 = a2, & if the circle is general then the equation of the polar becomesxx1 + yy1 + g (x + x1) + f (y + y1) + c = 0. Note that if the point (x1 , y1) be on the circle then thechord of contact, tangent & polar will be represented by the same equation.

(iii) Pole of a given line Ax + By + C = 0 w.r.t. any circle x2 + y2 = a2 is

C

aB,

C

aA 22

.



(iv) If the polar of a point P pass through a point Q, then the polar of Q passes through P.(v) Two lines L1 & L2 are conjugate of each other if Pole of L1 lies on L2 & vice versa Similarly two points

P & Q are said to be conjugate of each other if the polar of P passes through Q & vice-versa.

13. COMMON TANGENTS TO TWO CIRCLES :(i) Where the two circles neither intersect nor touch each other , there are FOUR common tangents,

two of them are transverse & the others are direct common tangents.(ii) When they intersect there are two common tangents, both of them being direct.(iii) When they touch each other :

(a) EXTERNALLY : there are three common tangents, two direct and one is the tangent at thepoint of contact .

(b) INTERNALLY : only one common tangent possible at their point of contact.(iv) Length of an external common tangent & internal common tangent to the two circles is given by:

Lext =2

212 )rr(d & Lint = 2

212 )rr(d .

Where d = distance between the centres of the two circles . r1 & r2 are the radii of the two circles.(v) The direct common tangents meet at a point which divides the line joining centre of circles

externally in the ratio of their radii.Transverse common tangents meet at a point which divides the line joining centre of circlesinternally in the ratio of their radii.

14. RADICAL AXIS & RADICAL CENTRE :The radical axis of two circles is the locus of points whose powers w.r.t. the two circles are equal. Theequation of radical axis of the two circles S1 = 0 & S2 = 0 is given ;S1 S2 = 0 i.e. 2 (g1 g2) x + 2 (f1 f2) y + (c1 c2) = 0.NOTE THAT :

(a) If two circles intersect, then the radical axis is the common chord of the two circles.(b) If two circles touch each other then the radical axis is the common tangent of the two circles at

the common point of contact.(c) Radical axis is always perpendicular to the line joining the centres of the two circles.(d) Radical axis need not always pass through the mid point of the line joining the centres of the two

circles.(e) Radical axis bisects a common tangent between the two circles.(f) The common point of intersection of the radical axes of three circles taken two at a time is

called the radical centre of three circles.(g) A system of circles , every two which have the same radical axis, is called a coaxial system.

(h) Pairs of circles which do not have radical axis are concentric.

15. ORTHOGONALITY OF TWO CIRCLES :Two circles S1= 0 & S2= 0 are said to be orthogonal or said to intersect orthogonally if the tangentsat their point of intersection include a right angle. The condition for two circles to be orthogonalis : 2 g1 g2 + 2 f1 f2 = c1 + c2 .

Note :(a) Locus of the centre of a variable circle orthogonal to two fixed circles is the radical axis between the

two fixed circles .(b) If two circles are orthogonal, then the polar of a point 'P' on first circle w.r.t. the second circle passes

through the point Q which is the other end of the diameter through P . Hence locus of a point whichmoves such that its polars w.r.t. the circles S1 = 0 , S2 = 0 & S3 = 0 are concurrent in a circle which isorthogonal to all the three circles.



SOLVED EXAMPLES

1. Find the equation of the circle whose centre is (1, –2) and radius is 4.Sol. The equation of the circle is (x – 1)2 + (y – (–2))2 = 42

(x – 1)2 + (y + 2)2 = 16 x2 + y2 – 2x + 4y – 11 = 0 Ans.

2. Find the equation of the circle which passes through the point of intersection of the lines 3x – 2y – 1 = 0and 4x + y – 27 = 0 and whose centre is (2, –3).

Sol. Let P be the point of intersection of the lines AB and LM whose equations are respectively3x – 2y – 1 = 0 .....(i)

and 4x + y – 27 = 0 .....(ii)Solving (i) and (ii) , we get x = 5, y = 7, So coordinates of P are (5, 7).Let C (2, –3) be the centre of the circle. Since the circle passes through P, therefore

CP = radius 2 2

5 2 7 3 = radius

radius = 109 .

Hence the equation of the required circle is (x – 2)2 + (y + 3)2 = 2

109

3. Find the centre & radius of the circle whose equation is x2 + y2 – 4x + 6y + 12 = 0.Sol. Comparing it with the general equation x2 + y2 + 2gx + 2fy + c = 0, We have

2g = – 4 g = – 22f = 6 f = 3& c = 12

centre is (–g, –f) i.e., (2, –3) and radius = 2 22 2g f c 2 3 12 1

4. Find the equation of the circle, the coordinates of the end points of whose diameter are (–1, 2) and(4, –3).

Sol. We know that the equation of the circle described on the line segment joining (x1, y1) and (x2, y2) asa diameter is (x – x1) (x – x2) + (y – y1) (y – y2) = 0.Here x1 = –1, x2 = 4, y1 = 2 and y2 = –3So, the equation of the required circle is(x + 1) (x – 4) + (y – 2) (y + 3) = 0 x2 + y2 – 3x + y – 10 = 0

5. Find the equation to the circle touching the y-axis at a distance –3 from the origin and intercepting alength 8 on the x-axis.

Sol. Let the equation of the circle be x2 + y2 + 2gx + 2fy + c = 0.Since it touches y-axis at (0, –3) and (0, –3) lies on the circle. c = f2 .....(i) 9 – 6f + c = 0 .....(ii)From (i) and (ii) , we get 9 – 6f + f2 = 0 (f – 3)2 = 0 f = 3.Putting f = 3 in (i) we obtain c = 9.It is given that the circle x2 + y2 + 2gx + 2fy + c = 0 intercepts length 8 on x-axis

22 g c 8 22 g 9 8 g2 – 9 = 16 g = ± 5

Hence, the required circle is x2 + y2 ± 10x + 6y + 9 = 0.



6. Find the parametric equations of the circle x2 + y2 – 4x – 2y + 1 = 0.Sol. We have : x2 + y2 – 4x – 2y + 1 = 0 (x2 – 4x) + (y2 – 2y) = –1

(x – 2)2 + (y – 1)2 = 22

So, the parametric equations of this circle arex = 2 + 2 cos , y = 1 + 2 sin .

7. Find the equations of the following curves in cartesian form. Also, find the centre and radius of the circlex = a + c cos , y = b + c sin

Sol. We have : x = a + c cos y = b + c sin cos = x a

c

, sin =

y b

c

2 2

x a y b

c c

= cos2 + sin2 (x – a)2 + (y – b)2 = c2

Clearly, it is a circle with centre at (a, b) and radius c.

8. Discuss the position of the points (1, 2) and (6, 0) with respect to the circle x2 + y2 – 4x + 2y – 11 = 0Sol. We have x2 + y2 – 4x + 2y – 11 = 0 or S = 0, where S = x2 + y2 – 4x + 2y – 11.

For the point (1, 2), we have S1 = 12 + 22 – 4 × 1 + 2 × 2 – 11 < 0For the point (6, 0), we have S2 = 62 + 02 – 4 × 6 + 2 × 0 – 11 > 0Hence, the point (1, 2) lies inside the circle and the point (6, 0) lies outside the circle.

9. For what value of c will the line y = 2x + c be a tangent to the circel x2 + y2 = 5 ?Sol. We have : y = 2x + c or 2x – y + c = 0..... (i) and x2 + y2 = 5 ..... (ii)

If the line (i) touches the circle (ii), thenLength of the from the centre (0, 0) = radius of circle (ii)

22

2 0 0 c5

2 1

c5

5

c

55 c = ± 5

Hence, the line (i) touches the circle (ii) for c = ± 5

10. Find the equation of the tangent to the circle x2 + y2 – 30x + 6y + 109 = 0 at (4, –1).Sol. Equation of tangent is

4x + (–y) – 30 y 1x 4

62 2

+ 109 = 0

or 4x – y – 15x – 60 + 3y – 3 + 109 = 0 or –11x + 2y + 46 = 0or 11x – 2y – 46 = 0Hence, the required equation of the tangent is 11x – 2y – 46 = 0.

11. Find the equation of tangents to the circle x2 + y2 – 6x + 4y – 12 = 0 which are parallel to the line4x + 3y + 5 = 0

Sol. Given circle is x2 + y2 – 6x + 4y – 12 = 0 .....(i)and given line is 4x + 3y + 5 = 0 .....(ii)Centre of circle (i) is (3, –2) and its radius is 5. Equation of any line4x + 3y + k = 0 parallel to the line (ii) .....(iii)



If line (iii) is tangent to circle, (i) then

2 2

| 4.3 3( 2) k |

4 3

= 5 or |6 + k| = 25

or 6 + k = ± 25 k = 19, – 31Hence equation of required tangents are 4x + 3y + 19 = 0 and 4x + 3y – 31 = 0

12. Find the equation of the normal to the circle x2 + y2 – 5x + 2y – 48 = 0 at the point (5, 6).Sol. The equation of the tangent to the circle x2 + y2 – 5x + 2y – 48 = 0 at (5, 6) is

5x + 6y – 5x 5 6

22 2

y

– 48 = 0 10x + 12y – 5x – 25 + 2y + 12 – 96 = 0

5x + 14y – 109 = 0

Slope of the tangent = 5

14 Slope of the normal =

14

5

Hence, the equation of the normal at (5, 6) isy – 6 = (14/5)(x – 5) 14x – 5y – 40 = 0

13. Find the equation of the pair of tangents drawn to the circle x2 + y2 – 2x + 4y = 0 from the point (0, 1)Sol. Given circle is S = x2 + y2 – 2x + 4y = 0 .....(i)

Let P (0, 1)For Point P, S1 = 02 + 12 – 2.0 + 4.1 = 5Clearly P lies outside the circleand T x . 0 + y . 1 – (x + 0) + 2(y + 1)i.e. T –x + 3y + 2.Now equation of pair of tangents from P(0, 1) to circle (1) is SS1 = T2

or 5(x2 + y2 – 2x + 4y) = (–x + 3y + 2)2

or 5x2 + 5y2 – 10x + 20y = x2 + 9y2 + 4 – 6xy – 4x + 12yor 4x2 – 4y2– 6x + 8y + 6xy – 4 = 0or 2x2 – 2y2 + 3xy – 3x + 4y – 2 = 0 ..... (ii)Note : Separate equation of pair of tangents : From (ii) 2x2 + 3(y – 1) x – (2y2 – 4y + 2) = 0

x = 2 23 y 1 9 y 1 8 2y 4y 2

4

or 4x + 3y – 3 = ± 225y 50y 25 = ± 5(y – 1)

Separate equations of tangents are x + 2y – 2 = 0 and 2x – y + 1 = 0.

14. Find the length of the tangent drawn from the point (5, 1) to the circle x2 + y2 + 6x – 4y – 3 = 0.Sol. Given circle is x2 + y2 + 6x – 4y – 3 = 0 .....(i)

Given point is (5, 1), Let P = (5, 1)

Now length of the tangent from P(5, 1) to circle (i) = 2 25 1 6.5 4.1 3 7

15. Find the equation of director circle of the circle (x – 2)2 + (y + 1)2 = 2.

Sol. Centre & radius of given circle are (2, –1) & 2 respectively..

Centre and radius of the director circle will be (2, –1) & 2 × 2 = 2 respectively..

equation of director circle is (x – 2)2 + (y + 1)2 = 4



x2 + y2 – 4x + 2y + 1 = 016. Find the equation of the chord of contact of the tangents drawn from (1, 2) to the circle

x2 + y2 – 2x + 4y + 7 = 0Sol. Given circle is x2 + y2 – 2x + 4y + 7 = 0 ..... (i)

Let P = (1, 2)For point P(1, 2), x2 + y2 – 2x + 4y + 7 = 1 + 4 – 2 + 8 + 7 = 18 > 0Hence point P lies outside the circleFor point P(1, 2), T = x . 1 + y . 2 – (x + 1) + 2(y + 2) + 7i.e., T = 4y + 10Now equation of the chord of contact of point P(1, 2) w.r.t. circle (i) will be4y + 10 = 0 or 2y + 5 = 0

17. Tangents are drawn to the circle x2 + y2 = 12 at the points where it is met by the circlex2 + y2 – 5x + 3y – 2 = 0 ; find the point of intersection of these tangents.

Sol. Given circles are S1 x2 + y2 – 12 = 0 .....(i)

and S2 = x2 + y2 – 5x + 3y – 2 = 0 .....(ii)Now equation of common chord of circle (i) and (ii) is

S1 – S2 = 0 i.e., 5x – 3y – 10 = 0Let this line meet circle (i) [or (ii) at A and B]Let the tangents to circle (i) at A and B meet at P(), then AB will be the chord of contact of thetangents to the circle (i) from P, therefore equation of AB will be

A

B

P

x+ y– 12 = 0Now lines (iii) and (iv) are same, therefore, equations (iii) and (iv) are identical

12

5 3 10

6, – 18

5

Hence P = 18

6,5

18. Find the equation of the polar of the point (2, –1) with respect to the circle x2 + y2 – 3x + 4y – 8 = 0.Sol. Given circle is x2 + y2 – 3x + 4y – 8 = 0 .....(i)

Given point is (2, –1) let P = (2, –1). Now equation of the polar of point P with respect to circle (i)

x.2 + y(–1) – 3x 2 y 1

42 2

– 8 = 0

or 4x – 2y – 3x – 6 + 4y – 4 – 16 = 0 or x + 2y – 26 = 0

19. Find the pole of the line 3x + 5y + 17 = 0 with respect to the circle x2 + y2 + 4x + 6y + 9 = 0.Sol. Given circle is x2 + y2 + 4x + 6y + 9 = 0 .....(i)

and given line is 3x + 5y + 17 = 0 .....(ii)Let P() be the pole of line (ii) with respect to circle (i)Now equation of polar of point P() with respect to circle (i) is

x+ y+ 2(x +) + 3(y + ) + 9 = 0or (+ 2)x + (+ 3)y + 2+ 3+ 9 = 0 .....(iii)



Now lines (ii) and (iii) are same, therefore,

2 3 2 3 9

3 5 17

(i) (ii) (iii)From (i) and (ii), we get

5+ 10 = 3 + 9 or 5 – 3= –1 .....(iv)From (i) and (iii), we get

17+ 34 = 6+ 9+ 27 or 11– 9= –7 .....(v)Solving (iv) & (v), we get = 1, = 2Hence required pole is (1, 2).

20. Find the equation of the chord of the circle x2 + y2 + 6x + 8y – 11 = 0, whose middle point is (1, –1).Sol. Equation of given circle is S x2 + y2 + 6x + 8y – 11 = 0

Let L (1, –1)For point L(1, –1), S1 = 12 + (–1)2 + 6.1 + 8(–1) – 11 = – 11 andT x.1 + y(–1) + 3(x + 1) + 4(y – 1) – 11 i.e., T 4x + 3y – 12Now equation of the chord of circle (i) whose middle point is L (1, –1) isT = S1 or 4x + 3y – 12 = – 11 or 4x + 3y – 1 = 0

Second Method

Let C be the centre of the given circle, then C (–3, –4), L (1, –1) slope of CL = 4 1 3

3 1 4

Equation of chord of circle whose middle point is L, is

y + 1 = –4

3(x – 1) [ chord is perpendicular to CL]

or 4x + 3y – 1 = 0

21. Examine if the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other externally orinternally.

Sol. Given circles are x2 + y2 – 2x – 4y = 0 .....(i)and x2 + y2 – 8y – 4 = 0 .....(ii)Let A and B be the centres and r1 and r2 the radii of circles (i) and (ii) respectively, then

A (1, 2), B (0, 4), r1 = 5 , r2 = 2 5

Now AB = 2 2

1 0 2 4 5 and r1+ r2 = 3 5 , |r1 – r2| = 5

Thus AB = |r1 – r2|, hence the two circles touch each other internally.

22. Obtain the equation of the circle orthogonal to both the circles x2 + y2 + 3x – 5y + 6 = 0 and4x2 + 4y2 – 28x + 29 = 0 and whose centre lies on the line 3x + 4y + 1 = 0.

Sol. Given circles are x2 + y2 + 3x – 5y + 6 = 0 .....(i)and 4x2 + 4y2 – 28x + 29 = 0

or x2 + y2 – 7x + 29

04 . .....(ii)

Let the required circle be x2 + y2 + 2gx + 2fy + c = 0 .....(iii)Since circle (iii) cuts circles (i) and (ii) orthogonally

2g3

2

+ 2f5

2

= c + 6 or 3g – 5f = c + 6 .....(iv)



and 2g7

2

+ 2f.0 = c + 29

4or –7g = c +

29

4.....(v)

From (iv) & (v) , we get 10g – 5f = 5

4

or 40g – 20f = – 5 .....(vi)Given line is 3x + 4y = – 1 .....(vii)Since centre (–g, – f) of circle (iii) lies on line (vii), –3g – 4g = –1 .....(viii)

Solving (vi) & (viii), we get g = 0, f = 1

4

from (5), c = 29

4 from (iii), required circle is

x2 + y2 + 1 29

y –2 4

= 0 or 4(x2 + y2) + 2y – 29 = 0

23. Find the co-ordinates of the point from which the lengths of the tangents to the following three circles beequal.

3x2 + 3y2 + 4x – 6y – 1 = 02x2 + 2y2 – 3x – 2y – 4 = 02x2 + 2y2 – x + y – 1 = 0

Sol. Here we have to find the radical centre of the three circles. First reduce them to standard form in whichcoefficients of x2 and y2 be each unity. Subtracting in pairs the three radical axis are

17 5x y 0

6 3 ;

3 3x y 0

2 2 ;

11 5 1x y 0

6 2 6

solving any two, we get the point 16 31

,21 63

which satisfies the third also. This point is called the

radical centre and by definition the length of the tangents from it to the three circles are equal.

24. Find the equations of the circles passing through the points of intersection of the circlesx2 + y2 – 2x – 4y – 4 = 0 and x2 + y2 – 10x – 12y + 40 = 0 and whose radius is 4.

Sol. Any circle through the intersection of given circles is S1 + S2 = 0or (x2 + y2 – 2x – 4y – 4) + (x2 + y2 – 10x – 12y + 40) = 0 .....(i)

or (x2 + y2) – 2(1 5 )

1

x – 2

(2 6 )

1

y +

40 4

1

= 0

r =2 2g f c = 4, given

16 =

2 2

2 2

1 5 2 6 40 4

11 1

16(1 + 2+ ) = 1 + 10+ 25+ 4 + 24 + 36– 40– 40+ 4 + 4or 16 + 32 + 16 = 21– 2+ 9 or 5– 34– 7 = 0 (– 7) (5+ 1) = 0 = 7, – 1/5Putting the values of in (i) the required circles are

2x2 + 2y2 – 18x – 22y + 69 = 0 and x2 + y2 – 2y – 15 = 0



25. Find the equations of circles which touche 2x – y + 3 = 0 and pass through the points of intersection ofthe line x + 2y – 1 = 0 and the circle x2 + y2 – 2x + 1= 0.

Sol. The required circle by S +L = 0 isx2 + y2 – 2x + 1 + (x + 2y – 1) = 0

or x2 + y2 – x(2 –) + 2y + (1 –) = 0centre (–g, –f) is [(2 – )/2, – ]

r = 2 2g f c

= 2 2 21

2 / 4 1 5 / 2 52

Since the circle touches the line 2x – y + 3 = 0 therefore perpendicular from centre is equal to radius

2.[(2 ) / ] ( ) 35

2

2

5

or 5 = ±

2

.5 = ± 2

Putting the values of in (i) the required circles arex2 + y2 + 4y – 1 = 0x2 + y2 – 4x – 4y + 3 = 0.

26. Find the equation of circle passing through the points A(1, 1) & B(2, 2) and whose radius is 1.Sol. Equation of AB is x – y = 0

equation of circle is(x – 1) (x – 2) + (y – 1) (y – 2) + (x – y) = 0

or x2 + y2 + (– 3) x – (+ 3) y + 4 = 0

radius =

2 23 3

44 4

But radius = 1 (given) ;

2 23 3

4 14 4

or (– 3)2 + (+ 3)2 – 16 = 4.or 2= 2or = ± 1 Equation of circle is

x2 + y2 – 2x – 4y + 4 = 0& x2 + y2 – 4x – 2y + 4 = 0 Ans.

27. Find the equation of the circle passing through the point (2, 1) and touching the line x + 2y – 1 = 0 at thepoint (3, – 1).

Sol. Equation of circle is(x – 3)2 + (y + 1)2 + (x + 2y – 1) = 0Since it passes through the point (2, 1)

1 + 4 + (2 + 2 – 1) = 0 = –5/3 circle is

(x – 3)2 + (y + 1)2 – 5/3 (x + 2y – 1) = 0 3x2 + 3y2 – 23x – 4y + 35 = 0 Ans.



28. Find the equation of circle circumscribing the triangle whose sides are 3x – y – 9 = 0, 5x – 3y – 23 = 0& x + y – 3 = 0.

Sol.

X

L : 3x – y –9 = 01 L : 5x –3y –23 = 02

L : x +y –3 = 03

A

B C

L1L2 + L2L3 + L1L3 = 0(3x – y – 9) (5x – 3y – 23) + (5x – 3y – 23) (x + y – 3) + (3x – y – 9)(x + y – 3) = 0(15x2 + 3y2 – 14xy – 114x + 50y + 207) + (5x2 – 3y2 + 2xy – 38x – 14y + 69)

+ (3x2 – y2 + 2xy – 18x – 6y + 27) = 0(5+ 3+ 15) x2 + (3 – 3–)y2 + xy (2+ 2– 14) – x (114 + 38+ 18)

+ y(50 – 14– 6) + (207 + 69+ 27) = 0 .....(i)coefficient of x2 = coefficient of y2

5+ 3+ 15 = 3 – 3– ++ 12 = 02+ + 3 = 0 .....(ii)

coefficient of xy = 0 2+ 2– 14 = 0 – 7 = 0 .....(iii)Solving (ii) and (iii) , we have

= – 10, = 17Putting these values of & in equation (i) , we get

2x2 + 2y2 – 5x + 11y – 3 = 0

29. Find the locus of the points of intersection of the tangents to the circle x = r cos , y = r sin at pointswhose parametric angles differ by /3.

Sol. All such points P satisfying the given condition will be equidistant from the origin O (see fig.)Hence the locus of P will be a circle centred at the origin, having radius equal to

OP = r 2r

3cos6

O

BP

A

Therefore, equation of the required locus is x2 + y2 = 24

r3

.

30. If – 3l2 – 6l – 1 + 6m2 = 0, find the equation of the circle for which lx + my + 1 = 0 is a tangent.Sol. The given expression can be written as

6(l2 + m2) = 9l2 + 6l + 1 i.e.,2 2

3 16

m

l

l

From this expression we can infer that the perpendicular distance of the point (3, 0) from the line

lx + my + 1 = 0 is 6 .

This proves that the given line is a tangent to the circle (x – 3)2 + y2 = 6.



31. Prove that x2 + y2 = a2 and (x – 2a)2 + y2 = a2 are two equal circles touching each other. Find theequation of circle (or circles ) of the same radius touching both the circles.

Sol. Given circles arex2 + y2 = a2 .....(1)

and (x – 2a)2 + y2 = a2 .....(2)Let A and B be the centres and r1 and r2 the radii of the circles (1) and (2) respectively. Then

A (0, 0), B (2a, 0), r1 = a, r2 = a

Now AB = 2 20 2a 0 = 2a = r1 + r2

Hence the two circles touch each other externally.Let the equation of the circle having same radius 'a' and touching the circles (1) and (2) be

(x – )2 + (y –)2 = a2 .....(3)Its centre C is () and radius r3 = aSince circle (3) touches the circle (1)

AC = r1 + r3 = 2a. [Here AC |r1 – r3| as r1 – r3 = a – a = 0] AC2 = 4a2�2 + = 4a

Again since circle (3) touches the circle (2)BC = r2 + r3

(2a –)2 + = (a + a)2

– 4a 4a2 – 4a = 0

= a we get = ± 3 a.

Hence, the required circles are (x – a)2 + (y a 3 )2 = a2

or x2 + y2 – 2ax 2 3 ay + 3a2 = 0

32. If the curves ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 and Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0 intersect

at four concyclic points, prove that a b A B

h H

.

Sol. Equation of a curve passing through the intersection points of the given curves can be written as(ax2 + 2hxy + by2 + 2gx + 2fy + c) + (Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C) = 0 .....(1)If this curve must be a circle, then coeff. of x2 = coeff. of y2

i.e., (a + A) = (b + B) gives = b a

A B

.....(2)

and coeff. of xy = 0

i.e., 2(h + H) = 0 given = – h

H.....(3)

Equating the two values of , we get the desired result.

33. Let S x2 + y2 + 2gx + 2fy + c = 0 be a given circle. Find the locus of the foot of the perpendiculardrawn from the origin upon any chord of S which subtends right angle at the origin.

A

B

O

P(h,k)



AB is a variable chord such that = AOB = 2

.

Let P(h, k) be the foot of the perpendicular drawn from origin upon AB. Equation of the chord AB is

y – k = h

k

(x – h)

i.e., hx + ky = h2 + k2 .....(1)Equation of the pair of straight lines passing through the origin and the intersection point of the givencircle

x2 + y2 + 2gx + 2fy + c = 0 .....(2)and the variable chord AB is

x2 + y2 + 2(gx + fy)2

2 2 2 2

hx ky hx kyc 0

h k h k

.....(3)

If equation (3) must represent a pair of perpendicular lines, then we have coeff. of x2 + coeff. of y2 = 0

i.e.,

2 2

2 22 2 2 22 2 2 2

2gh ch 2fk ck1 1 0

h k h kh k h k

Putting (x, y) in place of (h, k) gives the equation of the required locus as

x2 + y2 + gx + fy + c

2 = 0.

34. The line Ax + By + C = 0 cuts the circle x2 + y2 + gx + fy + c = 0 at P and Q.The line A'x + B'y + C' = 0 cuts the circle x2 + y2 + g'x + f 'y + c' = 0 at R and S.

If P, Q, S are concyclic, show that

g g ' f f ' c c '

A B C 0

A' B' C'

Sol. Equation of a circle through P and Q is x2 + y2 + gx + fy + c + (Ax + By + C) = 0i.e., x2 + y2 + (g + A) x + (f + B)y + (c + C) = 0 .....(1)and equation of a circle through R and S is x2 + y2 + g'x + f 'y + c' + (A'x + B'y + C') = 0

x2 + y2 + (g' + A') y + (f ' + B') + (c' + C') = 0 .....(2)If P, Q, R and S are concyclic points, then equations (1) and (2) must represent the same circle.

Equating the ratio of the coefficients, we have 1 = g A f B c C

g ' A ' f ' B' c ' C '

i.e A – ' + g – g' = 0 .....(3)B – B' + f – f' = 0 .....(4)

and C – C' + c – c' = 0 .....(5)

Eliminating and from equation (3), (4) and (5), we have

A A' g g '

B B' f f '

C C' c c '

= 0

or

g g ' f f ' c c '

A B C 0

A' B' C'

[Interchanging rows by columns and then interchanging the second and

the third row]



Aliter : Let the given circles be S1 x2 + y2 + gx + fy + c = 0 .....(1)and S2 x

2 +y2 + g'x + f 'y + c' = 0 .....(2)If S be the required circle, then according to the given conditionAx + By + C = 0 is the radical axis of S1, Sand A'x + B'y + C' = 0 is the radical axis of S2, Swhile (g – g')x + (f – f ')y + (c – c') = 0 is the radical axis of S1,S2.Since the radical axes of three circles taken in pairs are concurrent, therefore, we have

g g ' f f ' c c '

A B C 0

A' B' C'

which is the desired result.

35. Circles are drawn passing through the origin O to intersect the coordinate axes at point P and Q such thatm OP + n. OQ is a constant. Show that the circles pass through a fixed point.

Sol. Equation of a circle passing through the origin and having X and Y intercepts equal to a and b respectivelyis x2 + y2 – ax – by = 0 .....(1)According to the given condition, we havema + nb = k (constant)

i.e., k ma

bn

.....(2)

Putting the above value of b in equation (1) , we have x2 + y2 – ax – k ma

n

y = 0

i.e., {n(x2 + y2) – ky} – a (nx – my) = 0Which represents the equation of a family of circles passing through the intersection points of the circle

n (x2 + y2) – ky = 0 .....(3)and the line

nx – my = 0 .....(4)

Solving equation (3) and (4), gives the coordinates of the fixed point as 2 2 2 2

mk nk,

m n m n

.

36. P(p, q) is a point on a circle passing through the origin and centred at Cp q

,2 2

. If two distinct chords

can be drawn from P such that these chords are bisected by the X-axis, then show that p2 > 8q2.

Sol. It can be seen that the given points P(p, q), Cp q

,2 2

and the origin are collinear which implies that line

OP where O is the origin is a diameter of the given circle. Therefore, equation of the given circle isx(x – p) + y(y – q) = 0

i.e x2 + y2 – px – qy = 0 .....(1)Let M(a, 0) be the mid-point of a chord AP (see fig.) Then, we have

CM AP

O X

(p/2, q/2) CP(p,q)

A

Y

M(a,

0)

i.e., slope of CM × slope of AP = –1

qq2 1

p p aa2

i.e., q2 + (p – 2a) (p – a) = 0 i.e., 2a2 – 3pa + p2 + q2 = 0 .....(2)



Equation (2) which is a quadratic equation in a shows that there will be two real and distinct values of aif the discriminant is > 0i.e., if (3p)2 – 4 × 2(p2 + q2) > 0i.e., if p2 > 8q2

which is the desired result.

Aliter. Equation of the given circle isx2 + y2 – px – qy = 0 .....(1)

Equation of any line through P(p, q) can be written asy – q = m (x – p) (where m is a variable)

i.e., y mp q

xm

.....(2)

putting the value of x from equation (2) in equation (1) will give the ordinate of the intersection points of

the line and the given circle as

2

2y mp q y mp qy p qy 0

m m

i.e., {y + (mp – q)}2 + m2y2 – mp{y + (mp – q)} – m2qy = 0i.e., (1 + m2)y2 + {2(mp – q) – mp – m2q}y + (mp – q)2 – mp(mp – q) = 0i.e., (1 + m2)y2 + (pm – 2q – qm2)y – q(mp – q) = 0 .....(3)The above equation gives the Y coordinates of the intersection points of the chord and the given circle.According to the given condition, the mid-point of this intercept lies on the X-axis, therefore we havesum of the roots of equation (3) = 0i.e., pm – 2q – qm2 = 0i.e., qm2 – pm + 2q =0 .....(4)The above equation shows that there will be two real and distinct values of m if p2 > 8q2 which is thedesired result.

37. Prove that the square of the tangent that can be drawn from any point on one circle to another circle isequal to twice the product of perpendicular distance of the point from the radical axis of two circles anddistances between their centres.

Sol. Let us choose the circles, as S1 x2 + y2 – a2 = 0 .....(1)

and S2 (x – b)2 + y2 – c2 = 0 .....(2)Let P (a cos, a sin) be any point on circle S1. The length of the tangent from P to circle S2, is given byPT2 = S2(a cos , a sin ) = (acos – b)2 + (a sin )2 – c2 = a2 + b2 – c2 – 2ab cosThe distance between the centres of S1 and S2 isC1C2 = bThe radical axis of S1 and S2, is 2bx – a2 – b2 + c2 = 0[equation (1) – equation (2)]The perpendicular distance of P from the radical axis, is

PM = 2 2 2| 2b a cos a b c |

2b

Now, we have

2.PM. C1C2 = 2b. 2 2 2| 2abcos a b c |

2b

= |a2 + b2 – c2 – 2ab cos| = PT2 which proves the

desired result.



38. Consider a family of circles passing through the intersection point of the lines 3 (y – 1) = x – 1 and

y – 1 = 3 (x – 1) and having its centre on the acute angle bisector of the given lines. Show that the

common chords of each member of the family and the circle x2 + y2 + 4x – 6y + 5 = 0 are concurrent.Find the point of concurrency.

Sol. The given lines 3 (y – 1) = x – 1 .....(1)

y – 1 = 3 (x – 1) .....(2)

intersect at the point (1, 1)Rewriting the equation of the given lines such that their constant terms are both positive, we have

x – 3 y + 3 – 1 = 0 .....(3)

and – 3 x + y + 3 – 1 = 0 .....(4)

Here, we have

(product of coeff's of x) + (product of coeff's of y) = – 3 – 3 = – ve quantity which implies that the

acute angle between the given lines contains the origin.Therefore, equation of the acute angle bisector of the given lines is

x 3y 3 1 3x y 3 1

2 2

i.e., y = xAny point on the above bisector can be chose as () and equation of any circle passing through(1, 1) and having centre at (, ) is

(x –)2 + (y –)2 = (1 –)2 + (1 – )2 .....(6)i.e., x2 + y2 – 2x – 2y + 4– 2 = 0The common chord of the given circle

x2 + y2 + 4x – 6y + 5 = 0 .....(7)and the circle represented by equation (6) is

(4 + 2) x + (2– 6)y + (7 – 4) = 0i.e., (4x – 6y + 7) + 2(x + y – 2) = 0 .....(8)Which represents a family of straight lines passing through the intersection point of the lines

4x – 6y + 7 = 0 .....(9)and x + y – 2 = 0 .....(10)

Solving equation (9), (10) gives the coordinates of the fixed point as 1 3

,2 2

.

39. Find the range of value of for which the variable line 3x + 4y – = 0 lies between the circlesx2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 without intercepting a chord on either circle.

Sol. The given circleS1 x

2 + y2 – 2x – 2y + 1 = 0 .....(1)has centre C1 (1, 1) and radius r1 = 1

C1C2

M2

O

Y

M1

X

The other given circleS2 x

2 + y2 – 18x – 2y + 78 = 0 .....(2)has centre C2 (9, 1) and radius r2 = 2.According to the required condition, we have

C1M1 r1

i.e.,2 2

| 3 4 |1

3 4



i.e., ( – 7) 5 [C1 lies below the line (7 – ) is a –ve quantity]i.e., 12 i.e.,

2 2

| 27 4 |2

3 4

i.e., (31– ) 10 [ C2 lies below the line] (31 –) is a + ve quantity]i.e., 21Hence, the permissible values of are 12 21.

40. Point P having integral coordinates lies on x2 + y2 = 1 and x2 + y2 + 2x + 4y + 1 = 0. A chord through

P meets the two circles at A and B. Find the equation of the chord PAB if PA and PB subtend equal

angles at the centres of the respective circles.

Sol. Equation of the given circles are S1 x2 + y2 – 1 = 0 .....(1)

and S2 x2 + y2 + 2x + 4y + 1 = 0 .....(2)

Subtracting equation (2) from equation (1) we have

x = – (2y + 1) .....(3)

Putting in equation (1), we have (2y + 1)2 + y2 = 1

i.e., 5y2 + 4y = 0 gives y = 0, – 4/5

and the corresponding values of x = – 1, 3/5.

Thus the intersection point of circles S1 and S2, having integral coordinates, is P (–1, 0).

From the fig., we can see that if PA and PB subtend equal angles at C1 and C2 respectively, then

PA : PB = C1A : C2B = 1 : 2

Equation of a line through P can be chosen as

y = m (x + 1) .....(4)

Solving equations (1) and (4) for the intersection point () (say) , we have

x2 + m2(x + 1)2 = 1

i.e., (1 + m2) x2 + 2m2x + (m2 – 1) = 0

whose one root is x = –1 since one of the intersection point is P(–1, 0)

Thus, we have –1 × 2

2

m 1

1 m

gives

2

2

1 m

1 m

Solving equation (2) and (4) for the intersection point B(, ) (say), we have

x2 + m2(x + 1)2 + 2x + 4m(x + 1) + 1 = 0

i.e., (1 + m2)x2 + (2m2 + 4m + 2)x + (m2 + 4m + 1) = 0

whose one root is x = –1 since one of the intersection point is P(–1, 0).

Thus, we have –1 × = 2

2

m 4m 1

1 m

given =

2

2

m 4m 1

1 m

Now, using the condition PA : PB = 1 : 2, we have 2 + = –3

i.e., 2(1 – m2) – (m2 + 4m + 1) = –3(1 + m2)

gives m = 1

Hence, equation of the required chord is y = 1(x + 1).



41. Curves ax2 + 2hxy + by2 – 2gx – 2fy + c = 0 and a'x2 – 2hxy + (a' + a – b)y2 – 2g'x – 2f 'y + c = 0

intersect at four concyclic point A, B, C and D. If P is the point g ' g f ' f

,a ' a a ' a

prove that PAA2 + PB2

+ PC2 = 3PD2.Sol. Equation of a curve passing through the intersection points of the given curves

ax2 + 2hxy + by2 – 2gx – 2fy + c = 0 .....(1)and a'x2 – 2hxy + (a' + a – b)y2 – 2g'x – 2f 'y + c = 0 .....(2)can be written as {a'x2 – 2hxy + (a' + a – b)y2 – 2g'x – 2f 'y + c}

+ {ax2 + 2hxy + by2 – 2gx – 2fy + c} = 0i.e., (a' + a)x2 + 2h( – 1)xy + (a' + a – b + b)y2

–2(g' + g)x – 2(f ' + f)y + (1 + )c = 0 .....(3)According to the given condition equation (3) must represent a circle, therefore,we have coeff. of x2 = coeff. of y2

i.e., a' + a = a' + a – b + bi.e., (a – b) = a – bgives = 1 and coeff. of xy = 0i.e., – 1 = 0gives = 1The identical values prove that the curve is a circle.Putting the above value of in equation (3) gives the equation of the circle passing through the intersectionpoints of the curves represented by equation (1) and (2) as

(a' + a)(x2 + y2) – 2 (g' + g)x – 2(f ' + f)y + 2c = 0

which has its centre at the point g ' g f ' f

,a ' a a ' a

.

We can see that the conditions of the given point P is the same as the centre of the circle passing throughthe points A, B, C and D. Therefore, we have PA2 = PB2 = PC2 = PD2 = radius of the circle which givesthe desired result PA2 + PB2 + PC2 = 3PD2.

42. A is one of the points of intersection of two given circles. A variable line through A meets the two circlesagain at point P and Q. Show that the locus of the mid-point of P and Q is also a circle passing throughA.

Sol. Let us choose the intersection point A as the origin and the radical axis of the circles, as the Y-axis(see fig.). Then the equation of the circles can be chosen as

S1 x2 + y2 – 2g1x – 2fy = 0 .....(1)and S2 x2 + y2 – 2g2x – 2fy = 0 .....(2)Equation of a variable line through A can be written as y = mx.Putting in equation (1), we have

x2(1 + m2) – 2(g1 + mf)x = 0

gives x = 0, 1

2

2(g mf )

1 m

Putting in equation (2), we have x2(1 + m2) – 2(g2 + mf)x = 0

gives x = 0, 2

2

2(g mf )

1 m

Thus, we have P (x1, mx1) where x1 = 1

2

2(g mf )

1 m

and Q (x2, mx2) where x2 =

22

2(g mf )

1 m

If M(h, k) be the mid-point of PQ, then 2h = x1 + x2



i.e., h(1 + m2) = g1 + g2 + 2mf .....(3)and 2k = m(x1 + x2)i.e., k(1 + m2) = m(g1 + g2 + 2mf) .....(4)

Dividing equation (4) by equation (3), we have m = k

h

Putting the above value of m in equation (3), we have

2

2

kh 1

h

= g1 + g2 +

2kf

h i.e., h2 + k2 = (g1 + g2)h + 2fk

Putting (x, y) in place of (h, k) gives the equation of the required locus, asx2 + y2 – (g1 + g2)x – 2fy = 0

which is a circle passing through A(0, 0).

43. Q is a fixed point and S is a fixed circle. A variable chord through Q meets the circle S at point A and B.Find the locus of a point P on this chord such that QA, QP, QB are in(a) arithmetic progression(b) geometric progression(c) harmonic progression

Sol. Let us choose the line joining Q and the centre of the circle S as the X-axis and the centre of the circle asthe origin (see fig).

Let the coordinates of the fixed point Q be (, 0) and the equation of the fixed circle S bex2 + y2 = a2 .....(1)

Let be the inclination of a variable line through Q. The coordinates of any point on this line can bechosen as ( + r cos , r sin ). If this point also lies on the circle S, then putting the above coordinatesin equation (1),we have

( + r cos )2 + (r sin )2 = a2

i.e., r2 + (2 cos )r + 2 – a2 = 0 .....(2)The roots of the above equation, say r1, r2 are the distance QA and QB. Thus, we have

QA + QB = r1 + r2 = – 2 cos QA QB = r1r2 = 2 – a2

Let P(h, k) be the point whose locus is to be found. If the distance QP is denoted by r, then we haveh = + r cos , k = r sin

(a) If QP is the A.M. of QA and QB, then r = 1 2r r

2

– cos [from equation (2)]

Thus, we have h = – cos2 = sin2 .....(3)k = – cos sin .....(4)

Now, we have 2k

h = cos2 .....(5)



Adding equations (3) and (4), we have h2 + k2 = hHence, the required locus is x2 + y2 – x = 0 which is a circle.(b) If QP is the G.M. of QA and QB. then

r = 2 2

1 2r r a [from equation (2)]

Thus, we have h = – 2 2a cos , k = 2 2a sin

Eliminating , we have (h – )2 + k2 = 2 – a2

i.e., h2 + k2 – 2h + 2 = 0Hence, the required locus is x2 + y2 – 2x = 0 which is a circle.

(c) If QP is the H.M. of QA and QB, then r =

2 21 2

1 2

2r r a

r r cos

Thus, we have h = – 2 2a

, k =

2 2a

cos

sin

From the first equation above , is eliminated. h = a2

Hence, the required locus is x = a2.

44. Tangents are drawn to the circle x2 + y2 = 50 from a point 'P' lying on the x-axis. These tangents meet they-axis at points 'P1' and 'P2'. Possible coordinates of 'P' so that area of triangle PP1P2 is minimum, is/are

(a) (10, 0) (B) (10 2 , 0) (C) (–10, 0) (D) (–10 2 , 0)

Sol. OP = 5 2 sec ,

OP1 = 5 2 cosec ,

area (PP1P2) = 100

sin 2, area (PP1P2)min = 100

= /4 OP = 10 P = (10, 0), (–10, 0)Hence, (A), (C) are correct.

45. Two circles with radii 'r1' and 'r2', r1 > r2 2, touch each other externally. If '' be the angle between thedirect common tangents, then

(A) = sin–1 1 2

1 2

r r

r r

(B) = 2sin–1 1 2

1 2

r r

r r

(C) = sin–1 1 2

1 2

r r

r r

(D) none of these

Sol. sin = 1 2

1 2

r r

r r

= 2sin–1 1 2

1 2

r r

r r

Hence (B) is correct



46. If the curves ax2 + 4xy + 2y2 + x + y + 5 = 0 and ax2 + 6xy + 5y2 + 2x + 3y + 8 = 0 intersect at fourconcyclic points then the value of a is(A) 4 (B) –4 (C) 6 (D) –6

Sol. Any second degree curve passing through the intersection of the given curves isax2 + 4xy + 2y2 + x + y + 5 +(ax2 + 6xy + 5y2 + 2x + 3y + 8) = 0

If it is a circle, then coefficient of x2 = coefficient of y2 and coefficient of xy = 0a(1 + ) = 2 + 5 and 4 + 6 = 0

a = 2 5

1

and =

2

3 a =

102

32

13

= – 4.

Hence (B) is correct answer.

47. The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circlex2 + y2 = 1 pass through a fixed point -

(A) (2, 4) (B) 1 1

,2 4

(C) 1 1

,2 4

(D) (–2, –4)

Sol. The chord of contact of tangents from (, ) isx + y = 1 .....(1)

Hence, (1) passes through 1 1

,2 4

Hence (C) is correct answer.

48. Equation of chord AB of circle x2 + y2 = 2 passing through P(2, 2) such that PB/PA = 3, is given by -(A) x = 3y (B) x = y

(C) y – 2 = 3 (x – 2) (D) none of these

Sol. Any line passing through (2, 2) will be of the form y 2 x 2

sin cos

= r

When this line cuts the circle x2 + y2 = 2, (r cos + 2)2 + (2 + r sin )2 = 2 r2 + 4(sin + cos ) r + 6 = 0

2

1

PB r

PA r , now if r1 = , r2 = 3,

then 4 = –4(sin + cos ), 32 = 6 sin 2 = 1 = /4So required chord will be y – 2 = 1(x – 2) y = x.

Alternative solutionPA. PB = PT2 = 22 – 2 = 6 .....(1)

PB

PA = 3 .....(2)

From (1) and (2), we have PA = 2 , PB = 3 2

AB = 2 2 . Now diameter of the circle is 2 2 (as radius is 2 )Hence line passing through the centre y = x.Hence (B) is the correct answer.



49. Equation of a circle S(x, y) = 0, (S, (2, 3) = 16) which touches the line 3x + 4y – 7 = 0 at (1, 1) is givenby(A) x2 + y2 + x + 2y – 5 = 0 (B) x2 + y2 + 2x + 2y – 6 = 0(C) x2 + y2 + 4x – 6y = 0 (D) none of these

Sol. Any circle which touches 3x + 4y – 7 = 0 at (1, 1) will be of the formS(x, y) (x – 1)2 + (y – 1)2 + (3x + 4y – 7) = 0

Since, S(2, 3) = 16 = 1, so required circle will bex2 + y2 + x + 2y – 5 = 0Hence, (A) is the correct answer.

50. If (a, 0) is a point on a diameter of the circle x2 + y2 = 4, then x2 – 4x – a2 = 0 has(A) exactly one real root in (–1, 0] (B) exactly one real root in [2, 5](C) distinct roots greater than –1 (D) distinct roots less than 5

Sol. Since (a, 0) is a point on the diameter of the circle x2 + y2 = 4.So maximum value of a2 is 4.

Let f(x) = x2 – 4x – a2

Clearly f(–1) = 5 – a2 0 f(2) = – (a2 + 4) < 0f(0) = – a2 < 0 and f(5) = 5 – a2 > 0

So graph of f(x) will be as shownHence (A), (B), (C), (D) are the correct answer.

51. If a circle S(x, y) = 0 touches at the point (2, 3) of the line x + y = 5 and S(1, 2) = 0, then radius of suchcircle

(A) 2 units (B) 4 units (C) 1

2 units (D)

1

2 units

Sol. Desired equation of the circle is(x – 2)2 + (y – 3)2 + (x + y – 5) = 01 + 1 + (1 + 2 – 5) = 0 = 1x2 – 4x + 4 + y2 – 6y + 9 + x + y – 5 = 0 x2 + y2 – 3x – 5y + 8 = 0

2 2

x3 5

y2 2

= – 8 +

25 9 2 1

4 4 4 2

Hence (D) is the correct answer.

52. If P(2, 8) is an interior point of a circle x2 + y2 – 2x + 4y – p = 0 which neither touches nor intersects theaxes, then set for p is -(A) p < –1 (B) P < –4 (C) p > 96 (D)

Sol. For internal point p(2, 8), 4 + 64 – 4 + 32 – p < 0 p > 96 and x intercept = 2 1 p therefore

1 + p < 0

p < –1 and y intercept2 4 p p < – 4

Hence (D) is the correct answer.

53. If two circles (x –1 )2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct point then(A) 2 < r < 8 (B) r < 2 (C) r = 2 (D) r > 2

Sol. Let d be the distance between the centres of two circles of radii r1 and r2.These circle intersect at two distinct points if |r1 – r2| < d < r1 + r2



Here, the radii of the two circles are r and 3 and distance between the centres is 5.Thus, |r – 3| < 5 < r + 3 –2 < r < 8 and r > 2 2 < r < 8Hence (A) is the correct answer.

54. The common chord of x2 + y2 – 4x – 4y = 0 and x2 + y2 = 16 subtends at the origin an angle equal to(A) /6 (B) /4 (C) /3 (D) /2

Sol. The equation of the common chord of the circles x2 + y2 – 4x – 4y = 0 and x2 + y2 = 16 is x + y = 4

which meets the circle x2 + y2 = 16 at points A(4, 0) and B(0, 4). Obviously OAOB.Hence the common chord AB makes a right angle at the centre of the circle x2 + y2 = 16.Hence (D) is the correct answer.

55. The number of common tangents that can be drawn to the circle x2 + y2 – 4x – 6y – 3 = 0 andx2 + y2 + 2x + 2y + 1 = 0 is(A) 1 (B) 2 (C) 3 (D) 4

Sol. The two circles arex2 + y2 – 4x – 6y – 3 = 0 and x2 + y2 + 2x + 2y + 1 = 0Centre : C1 (2, 3), C2 (–1, –1) radii : r1 = 4, r2 = 1We have C1C2 = 5 = r1 + r2, therefore there are 3 common tangents to the given circles.Hence (C) is the correct answer.

56. The tangents drawn from the origin to the circle x2 + y2 – 2rx – 2hy + h2 = 0 are perpendicular if(A) h = r (B) h = –r (C) r2 + h2 = 1 (D) r2 = h2

Sol. The combined equation of the tangents drawn from (0, 0) tox2 + y2 – 2rx – 2hy + h2 = 0 is(x2 + y2 – 2rx – 2hy + h2)h2 = (–rx – hy + h2)2

This equation represents a pair of perpendicular straight lines if coeff. of x2 + coeff. of y2 = 0i.e., 2h2 – r2 – h2 = 0 r2 = h2 or r = ± h. Hence (A), (B) and (D) are correct answers.

57. The equation(s) of the tangent at the point (0, 0) to the circle, making intercepts of length 2a and 2b unitson the coordinate axes, is(are) -(A) ax + by = 0 (B) ax – by = 0 (C) x = y (D) none of these

Sol. Equation of circle passing through origin and cutting off intercepts 2a and 2b units on the coordinate axesis x2 + y2 ± 2ax ± 2by = 0Hence, (A), (B) are correct answers.



ELEMENTARY EXERCISES

( SL. LONEY - EX-17 )

Find the equation to the circle

Q.1 Whose radius is 3 and whose centre is (–1, 2).

Q.2 Whose radius is 10 and whose centre is (–5, –6).

Q.3 Whose radius is a + b and whose centre is (a, – b).

Q.4 Whose radius is 2 2a b and whose centre is (– a, – b).

Find the coordinates of the centres and the radii of the circles whose equations are

Q.5 x2 + y2 – 4x – 8y = 41

Q.6 3x2 + 3y2 – 5x – 6y + 4 = 0

Q.7 x2 + y2 = k(x + k)

Q.8 x2 + y2 = 2gx – 2fy

Q.9 2 2 21 m (x y ) 2cx 2mcy 0

Draw the circles whose equations are

Q.10 x2 + y2 = 2ay

Q.11 3x2 + 3y2 = 4x

Q.12 5x2 + 5y2 = 2x + 3y

Q.13 Find the equation to the circle which passes through the points (1, –2) and (4, –3) and which has its

centre on the straight line 3x + 4y = 7.

Q.14 Find the equation to the circle passing through the points (0, a) and (b, h), and having its centre on the

axis of x.

Find the equations to the circles which pass through the points

Q.15 (0, 0), (a, 0) and (0, b)

Q.16 (1, 2), (3, –4) and (5, –6)

Q.17 (1, 1), (2, –1) and (3, 2)

Q.18 (5, 7), (8, 1) and (1, 3)

Q.19 (a, b), (a, – b) and (a + b, a – b)



Q.20 ABCD is a square whose side is a; taking AB and AD as axes, prove that the equation to the circle

circumscribing the square is x2 + y2 = a (x + y)

Q.21 Find the equation to the circle which passes through the origin and cuts off intercepts equal to 3 and 4

from the axes.

Q.22 Find the equation to the circle passing through the origin and the points (a, b) and (b, a). Find the lengths

of the chords that it cuts off from the axes.

Q.23 Find the equation to the circle which goes through the origin and cuts off intercepts equal to h and k from

the positive parts of the axes.

Q.24 Find the equation to the circle, of radius a, which passes through the two points on the axis of x which are

at a distance b from the origin.

Find the equation to the circle which

Q.25 touches both axis at a distance 5 from the origin.

Q.26 touches both axis and is of radius a.

Q.27 touches both axes and passes through the point ( –2, –3).

Q.28 touches the axis of x and passes through the two points (1, –2) and (3, –4).

Q.29 touches the axis of y at the origin and passes through the point (b, c).

Q.30 touches the axis of x at a distance 3 from the origin and intercepts a distance 6 on the axis of y.

Q.31 Points (1, 0) and (2, 0) are taken on the axis of x, the axes being rectangular. On the line joining these

points an equilateral triangle is described, its vertex being in the positive quadrant. Find the equations to

the circles described on its sides as diameters.

Q.32 If y = mx be the equation of a chord of a circle whose radius is a, the origin of coordinates being one

extremity of the chord and the axis of x being a diameter of the circle, prove that the equation of a circle

of which this chord is the diameter is (1 + m2) (x2 + y2) – 2a (x + my) = 0.

Q.33 Find the equation to the circle passing through the points (12, 43), (18, 39), and (42, 3) and prove that

it also passes through the points (–54, –69) and (–81, –38).

Q.34 Find the equation to the circle circumscribing the quadrilateral formed by the straight lines

2x + 3y = 2, 3x – 2y = 4, x + 2y = 3 and 2x – y = 3.

Q.35 Prove that the equation to the circle of which the points (x1 , y1) and (x2, y2) are the ends of a chord of

a segment containing an angle is

(x – x1) (x – x2) + (y – y1) (y – y2) ± cot [(x – x1) (y – y2) – (x – x2) (y – y1)] = 0.

Q.36 Find the equations to the circles in which the line joining the points (a, b) and (b, – a) is a chord subtending

an angle of 45° at any point on its circumference.



( SL. LONEY - EX-18)

Write down the equation of the tangent to the circle

Q.1 x2 + y2 – 3x + 10y = 15 at the point (4, –11).

Q.2 4x2 + 4y2 – 16x + 24y = 117 at the point 11

4,2

.

Find the equations to the tangents to the circle

Q.3 x2 + y2 = 4 which are parallel to the line x + 2y + 3 = 0

Q.4 x2 + y2 + 2gx + 2fy + c = 0 which are parallel to the line x + 2y – 6 = 0.

Q.5 Prove that the straight line y x c 2 touches the circle x2 + y2 = c2, and find its point of contact.

Q.6 Find the condition that the straight line cx – by + b2 = 0 may touch the circle x2 + y2 = ax + by and find

the point of contact.

Q.7 Find whether the straight line x y 2 2 , touches the circle

x2 + y2 – 2x – 2y + l = 0.

Q.8 Find the condition that the straight line 3x + 4y = k may touch the circle x2 + y2 = 10x.

Q.9 Find the value of p so that the straight line

x cos + y sin – p = 0

may touch the circle

x2 + y2 – 2ax cos – 2by sin – a2 sin2 = 0.

Q.10 Find the condition that the straight line Ax + By + C = 0 may touch the circle

(x – a)2 + (y – b)2 = c2

Q.11 Find the equation to the tangent to the circle x2 + y2 = a2 which

(i) is parallel to the straight line y = mx + c,

(ii) is perpendicular to the straight line y = mx + c,

(iii) passes through the point (b, 0), and

(iv) makes with the axes a triangle whose area is a2.

Q.12 Find the length of the chord joining the points in which the straight line x y

1a b , meet the circle

x2 + y2 = r2



Q.13 Find the equation to the circles which pass through the origin and cut off equal chords a from the straight

lines y = x and y = –x.

Q.14 Find the equation to the straight lines joining the origin to the points in which the straight line y = rnx + c

cuts the circle

x2 + y2 = 2ax + 2by

Hence find the condition that these points may subtend a right angle at the origin.

Find also the condition that the straight line may touch the circle.

Find the equation to the circle which

Q.15 has its centre at the point (3, 4) and touches the straight line 5x + 12y = 1.

Q.16 touches the axes of coordinates and also the line x y

1a b , the centre being in the positive quadrant.

Q.17 has its centre at the point (1, –3) and touches the straight line 2x – y – 4 = 0.

Q.18 Find the general equation of a circle referred to two perpendicular tangents as axes.

Q.19 Find the equation to a circle of radius r which touches the axis of y at a point distant h from the origin,

the centre of the circle being in the positive quadrant.

Prove also that the equation to the other tangent which passes through the origin is

(r2 – h2) x + 2rhy = 0

Q.20 Find the equation to the circle whose centre is at the point () and which passes through the origin,

and prove that the equation of the tangent at the origin is x + y = 0

Q.21 Two circles are drawn through the points (a, 5a) and (4a, a) to touch the axis of y. Prove that they

intersect at an angle 1 40

tan9

.

Q.22 A circle passes through the points ( –1, 1), (0, 6), and (5, 5). Find the points on this circle the tangents

at which are parallel to the straight line joining the origin to its centre.



( SL. LONEY - EX-19 )

Find the polar of the point

Q.1 (1, 2) with respect to the circle x2 + y2 = 7.

Q.2 (4, –1) with respect to the circle 2x2 + 2y2 = 11.

Q.3 (–2, 3) with respect to the circle x2 + y2 – 4x – 6y + 5 = 0.

Q.41

5,2

with respect to the circle 3x2 + 3y2 – 7x + 8y – 9 = 0.

Q.5 (a, –b) with respect to the circle x2 + y2 + 2ax – 2by + a2 – b2 = 0.

Find the pole of the straight line

Q.6 x + 2y = 1 with respect to the circle x2 + y2 = 5.

Q.7 2x – y = 6 with respect to the circle 5x2 + 5y2 = 9.

Q.8 2x + y + 12 = 0 with respect to the circle x2 + y2 – 4x + 3y – 1 = 0.

Q.9 48x – 54y + 53 = 0 with respect to the circle 3x2 + 3y2 + 5x – 7y + 2 = 0.

Q.10 ax + by + 3a2 + 3b2 = 0 with respect to the circle x2 + y2 + 2ax + 2by = a2 + b2.

Q.11 Tangents are drawn to the circle x2 + y2 = 12 at the points where it is met by the circle

x2 + y2 – 5x + 3y – 2 = 0;

find the point of intersection of these tangents.

Q.12 Find the equation to that chord of the circle x2 + y2 = 81 which is bisected at the point (–2, 3), and its

pole with respect to the circle.

Q.13 Prove that the polars of the point (1, –2) with respect to the circles whose equations are

x2 + y2 + 6y + 5 = 0 and x2 + y2 + 2x + 8y + 5 = 0

coincide; prove also that there is another point the polars of which with respect to these circles are the

same and find its coordinates.

Q.14 Find the condition that the chord of contact of tangents from the point (x', y') to the circle x2 + y2 = a2

should subtend a right angle at the centre.

Q.15 Prove that the distances of two points, P and Q, each from the polar of the other with respect to a circle,

are to one another inversely as the distances of the points from the centre of the circle.

Q.16 Prove that the polar of a given point with respect to any one of the circles x2 + y2 – 2kx + c2 = 0, where

k is variable, always passes through a fixed point, whatever be the value of k.



Q.17 Tangents are drawn from the point (h, k) to the circle x2 + y2 = a2; prove that the area of the triangle

formed by them and the straight line joining their points of contact is 2 2 2 3/2

2 2

a(h k a )

h k

.

Find the lengths of the tangents drawn

Q.18 to the circle 2x2 + 2y2 = 3 from the point (–2, 3).

Q.19 to the circle 3x2 + 3y2 – 7x – 6y = 12 from the point (6, –7).

Q.20 to the circle x2 + y2 + 2bx – 3b2 = 0 from the point (a + b, a – b).

Q.21 Given the three circles

x2 + y2 – 16x + 60 = 0,

3x2 + 3y2 – 36x + 81 = 0,

and x2 + y2 – 16x – 12y + 84 = 0,

find (1) the point from which the tangents to them are equal in length, and (2) this length.

Q.22 The distances from the origin of the centres of three circles x2 + y2 – 2x = c2 (where c is a constant and

a variable) are in geometrical progression; prove that the lengths of the tangents drawn to them from

any point on the circle x2 + y2 = c2 are also in geometrical progression.

Q.23 Find the equation to the pair of tangents drawn

(1) from the point (11, 3) to the circle x2 + y2 = 65,

(2) from the point (4, 5) to the circle 2x2 + 2y2 – 8x + 12y + 21 = 0



EXERCISE–IQ.1 Determine the nature of the quadrilateral formed by four lines 3x + 4y – 5 = 0; 4x – 3y – 5 = 0; 3x + 4y + 5 = 0

and 4x – 3y + 5 = 0. Find the equation of the circle inscribed and circumscribing this quadrilateral.

Q.2 A circle S = 0 is drawn with its centre at (–1, 1) so as to touch the circle x2 + y2 – 4x + 6y – 3 = 0externally. Find the intercept made by the circle S = 0 on the coordinate axes.

Q.3 The line lx + my + n = 0 intersects the curve ax2 + 2hxy + by2 = 1 at the point P and Q. The circle on PQas diameter passes through the origin. Prove that n2(a + b) = l2 + m2.

Q.4 One of the diameters of the circle circumscribing the rectangle ABCD is 4y = x + 7. If A & B arethe points (–3, 4) & (5,4) respectively, then find the area of the rectangle.

Q.5 Let L1 be a straight line through the origin and L2 be the straight line x + y = 1 . If the intercepts madeby the circle x2 + y2 x + 3y = 0 on L1 & L2 are equal, then find the equation(s) which represent L1.

Q.6 A circle passes through the points (–1, 1), (0, 6) and (5, 5). Find the points on the circle the tangents atwhich are parallel to the straight line joining origin to the centre.

Q.7 Find the equations of straight lines which pass through the intersection of the lines x 2y 5 = 0,7x + y = 50 & divide the circumference of the circle x2 + y2 = 100 into two arcs whose lengths arein the ratio 2 : 1.

Q.8 In the given figure, the circle x2 + y2 = 25 intersects the x-axis atthe point A and B. The line x = 11 intersects the x-axis at thepoint C. Point P moves along the line x = 11 above the x-axisand AP intersects the circle at Q. Find

(i) The coordinates of the point P if the triangle AQB has the maximum area.(ii) The coordinates of the point P if Q is the middle point of AP.(iii) The coordinates of P if the area of the triangle AQB is (1/4)th of the area of the triangle APC.

Q.9 A circle is drawn with its centre on the line x + y = 2 to touch the line 4x – 3y + 4 = 0 and pass throughthe point (0, 1). Find its equation.

Q.10 A point moving around circle (x + 4)2 + (y + 2)2 = 25 with centre C broke away from it either at the pointA or point B on the circle and moved along a tangent to the circle passing through the point D (3, – 3).Find the following.

(i) Equation of the tangents at A and B.(ii) Coordinates of the points A and B.(iii) Angle ADB and the maximum and minimum distances of the point D from the circle.(iv) Area of quadrilateral ADBC and the DAB.(v) Equation of the circle circumscribing the DAB and also the intercepts made by this circle on the

coordinate axes.

Q.11 Find the locus of the mid point of the chord of a circle x2 + y2 = 4 such that the segment intercepted bythe chord on the curve x2 – 2x – 2y = 0 subtends a right angle at the origin.

Q.12 Find the equation of a line with gradient 1 such that the two circles x2 + y2 = 4 andx2 + y2 – 10x – 14y + 65 = 0 intercept equal length on it.

Q.13 Find the locus of the middle points of portions of the tangents to the circle x2 + y2 = a2 terminated by thecoordinate axes.

Q.14 Tangents are drawn to the concentric circles x2 + y2 = a2 and x2 + y2 = b2 at right angle to one another.Show that the locus of their point of intersection is a 3rd concentric circle. Find its radius.

Q.15 Find the equation to the circle which is such that the length of the tangents to it from the points (1, 0),

(2, 0) and (3, 2) are 1, 7 , 2 respectively..



Q.16 Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radiusunity and centres (–3, 0), (–1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle Stouches the circle B and passes through the centre of the circle C. If the length of this chord can beexpressed as x , find x.

Q.17 If the variable line 3x – 4y + k = 0 lies between the circles x2 + y2 – 2x – 2y + 1 = 0 andx2 + y2 – 16x – 2y + 61 = 0 without intersecting or touching either circle, then the range of k is (a, b)where a, b I. Find the value of (b – a).

Q.18 Obtain the equations of the straight lines passing through the point A(2, 0) & making 45° angle with thetangent at A to the circle (x + 2)2 + (y 3)2 = 25. Find the equations of the circles each of radius 3whose centres are on these straight lines at a distance of 5 2 from A.

Q.19 A variable circle passes through the point A (a, b) & touches the x-axis; show that the locus of the otherend of the diameter through A is (x a)2 = 4by.

Q.20 Find the locus of the mid point of all chords of the circle x2 + y2 2x 2y = 0 such that the pair of linesjoining (0, 0) & the point of intersection of the chords with the circles make equal angle with axis of x.

Q.21 A circle with center in the first quadrant is tangent to y = x + 10, y = x – 6, and the y-axis. Let (h, k) be

the center of the circle. If the value of (h + k) = a + ab where a is a surd, find the value of a + b.

Q.22 A circle C is tangent to the x and y axis in the first quadrant at the points P and Q respectively. BC andAD are parallel tangents to the circle with slope – 1. If the points A and B are on the y-axis while C and

D are on the x-axis and the area of the figure ABCD is 900 2 sq. units then find the radius of the circle.

Q.23 Circles C1 and C2 are externally tangent and they are both internally tangent to the circle C3. The radii ofC1 and C2 are 4 and 10, respectively and the centres of the three circles are collinear. A chord of C3 is

also a common internal tangent of C1 and C2. Given that the length of the chord is p

nm where m, n

and p are positive integers, m and p are relatively prime and n is not divisible by the square of anyprime, find the value of (m + n + p).

Q.24 Find the equation of the circle passing through the three points (4, 7), (5, 6) and (1, 8). Also find thecoordinates of the point of intersection of the tangents to the circle at the points where it is cut by thestraight line 5x + y + 17 = 0.

Q.25 The line 2x – 3y + 1 = 0 is tangent to a circle S = 0 at (1, 1). If the radius of the circle is 13 . Find theequation of the circle S.

Q.26 Find the equation of the circle which passes through the point (1, 1) & which touches the circlex2 + y2 + 4x 6y 3 = 0 at the point (2, 3) on it.

Q.27 Find the equation of the circle whose radius is 3 and which touches the circle x2 + y2 – 4x – 6y – 12=0internally at the point (–1, – 1).

Q.28 Given that a right angled trapezium has an inscribed circle. Prove that the length of the right angled leg isthe Harmonic mean of the lengths of bases.

Q.29 Let K denotes the square of the diameter of the circle whose diameter is the common chord of thetwo circles x2 + y2 + 2x + 3y + 1 = 0 and x2 + y2 + 4x + 3y + 2 = 0

and W denotes the sum of the abscissa and ordinates of a point P where all variable chords of thecurve y2 = 8x subtending right angles at the origin, are concurrent.

and H denotes the square of the length of the tangent from the point (3, 0) on the circle2x2 + 2y2 + 5y –16 = 0.

Find the value of KWH.

Q.30 Let S1 = 0 and S2 = 0 be two circles intersecting at P (6, 4) and both are tangent to x-axis and line y = mx

(where m > 0). If product of radii of the circles S1 = 0 and S2 = 0 is 3

52, then find the value of m.



EXERCISE–II

Q.1 Show that the equation of a straight line meeting the circle x2 + y2 = a2 in two points at equal distances

'd' from a point (x1 , y1) on its circumference is xx1 + yy1 a2 + 2d2 = 0.

Q.2 A rhombus ABCD has sides of length 10. A circle with centre 'A' passes through C (the opposite vertex)likewise, a circle with centre B passes through D. If the two circles are tangent to each other, find thearea of the rhombus.

Q.3 Let A, B, C be real numbers such that(i) (sin A, cos B) lies on a unit circle centred at origin.(ii) tan C and cot C are defined.If the minimum value of (tan C – sin A)2 + (cot C – cos B)2 is a + 2b where a, b I, find the valueof a3 + b3.

Q.4 An isosceles right angled triangle whose sides are 1, 1, 2 lies entirely in the first quadrant with the

ends of the hypotenuse on the coordinate axes. If it slides prove that the locus of its centroid is

(3x y)2 + (x 3y)2 = 9

32.

Q.5 Real number x, y satisfies x2 + y2 = 1. If the maximum and minimum value of the expression x7

y4z

are M and m respectively, then find the value (2M + 6m).

Q.6 The radical axis of the circles x2 + y2 + 2gx + 2fy + c = 0 and 2x2 + 2y2 + 3x + 8y + 2c = 0 touchesthe circle x² + y² + 2x 2y + 1 = 0. Show that either g = 3/4 or f = 2.

Q.7 Find the equation of the circle through the points of intersection of circles x2 + y2 4x 6y 12=0and x2 + y2 + 6x + 4y 12 = 0 & cutting the circle x2 + y2 2x 4 = 0 orthogonally.

Q.8 The centre of the circle S = 0 lie on the line 2x 2y + 9 = 0 & S = 0 cuts orthogonally the circlex2 + y2 = 4. Show that circle S = 0 passes through two fixed points & find their coordinates.

Q.9(a) Find the equation of a circle passing through the origin if the line pair, xy – 3x + 2y – 6 = 0 is orthogonalto it. If this circle is orthogonal to the circle x2 + y2 – kx + 2ky – 8=0 then find the value of k.

(b) Find the equation of the circle which cuts the circle x2 + y2 – 14x – 8y + 64 = 0 and the coordinate axes orthogonally.

Q.10 Show that the locus of the centres of a circle which cuts two given circles orthogonally is a straight line& hence deduce the locus of the centers of the circles which cut the circles x2 + y2 + 4x 6y + 9=0 &x2 + y2 5x + 4y + 2 = 0 orthogonally. Interpret the locus.

Q.11 Find the equation of a circle which touches the line x + y = 5 at the point (2, 7) and cuts the circlex2 + y2 + 4x 6y + 9 = 0 orthogonally.

Q.12 Find the equation of the circle passing through the point (–6 , 0) if the power of the point (1, 1) w.r.t. thecircle is 5 and it cuts the circle x2 + y2 – 4x – 6y – 3 = 0 orthogonally.

Q.13 Consider a family of circles passing through two fixed points A (3, 7) & B(6, 5). Then the chords inwhich the circle x2 + y2 – 4x – 6y – 3 = 0 cuts the members of the family are concurrent at a point.Find the coordinates of this point.

Q.14 Find the equation of circle passing through (1, 1) belonging to the system of coaxal circles that aretangent at (2, 2) to the locus of the point of intersection of mutually perpendicular tangent to the circlex2 + y2 = 4.

Q.15 The circle C : x2 + y2 + kx + (1 + k)y – (k + 1) = 0 passes through two fixed points for every real numberk. Find(i) the coordinates of these two points. (ii) the minimum value of the radius of a circle C.



Q.16 Find the equation of a circle which is co-axial with circles 2x2 + 2y2 2x + 6y 3 = 0 &x2 + y2 + 4x + 2y + 1 = 0. It is given that the centre of the circle to be determined lies on the radical axisof these two circles.

Q.17 The circles, which cut the family of circles passing through the fixed points A (2, 1) and B (4, 3)orthogonally, pass through two fixed points (x1 , y1) and (x2 , y2), which may be real or imaginary. Find

the value of 32

31

32

31 yyxx .

Q.18 Find the equation of a circle which touches the lines 7x2 – 18xy + 7y2 = 0 and the circlex2 + y2 – 8x – 8y = 0 and is contained in the given circle.

Q.19 Find the equation of the circle which passes through the origin, meets the x-axis orthogonally & cuts thecircle x2 + y2 = a2 at an angle of 45º.

Q.20 Consider two circles C1 of radius 'a' and C2 of radius 'b' (b > a) both lying in the first quadrant and

touching the coordinate axes. In each of the conditions listed in column-I, the ratio of ab is given in

column-II.Column-I Column-II

(A) C1 and C2 touch each other (P) 22(B) C1 and C2 are orthogonal (Q) 3

(C) C1 and C2 intersect so that the common chord is longest (R) 32

(D) C2 passes through the centre of C1 (S) 223

(T) 223

EXERCISE–III

Q. 1 (a) If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where

pq q) are bisected by the x-axis, then

(A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D) p2 > 8q2

(b) Let L, be a striaght line through the origin and L2 be the straight line x + y = 1. If the intercepts

made by the circle x2 + y2 – x + 3y = 0, on L1 & L2 are equal, then which of the following

equations can represent L1 ?

(A) x + y = 0 (B) x – y = 0 (C) x + 7y = 0 (D) x – 7y = 0

(c) Let T1 , T2 be two tangents drawn from (–2, 0) onto the circle C : x2 + y2 = 1. Determine the

circles touching C and having T1 , T2 as their pair of tangents. Further, find the equations of all

possible common tangents to these circles, when taken two at a time.[JEE '99, 2+3+10]

Q.2 (a) The triangle PQR is inscribed in the circle, x2 + y2 = 25. If Q and R have co-ordinates (3, 4) &( 4, 3) respectively, then QPR is equal to(A) /2 (B) /3 (C) /4 (D) /6

(b) If the circles, x2 + y2 + 2 x + 2 k y + 6 = 0 & x2 + y2 + 2 k y + k = 0 intersect orthogonally,

then ' k ' is :(A) 2 or 3/2 (B) 2 or 3/2 (C) 2 or 3/2 (D) 2 or 3/2

[JEE '2000 (Screening), 1+1]

Q.3 (a) Extremities of a diagonal of a rectangle are (0, 0) & (4, 3). Find the equation of the tangents tothe circumcircle of a rectangle which are parallel to this diagonal.

(b) Find the point on the straight line, y = 2 x + 11 which is nearest to the circle,16(x2 + y2) + 32x 8y 50 = 0.



(c) A circle of radius 2 units rolls on the outerside of the circle, x2 + y2 + 4 x = 0 , touching itexternally. Find the locus of the centre of this outer circle. Also find the equations of the commontangents of the two circles when the line joining the centres of the two circles is inclined at anangle of 60º with x-axis. [REE '2000 (Mains) 3 + 3 + 5]

Q.4 (a) Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS andRQ intersect at a point X on the circumference of the circle then 2r equals

(A) PQ RS (B) PQ RS

2 (C)

2PQ RS

PQ RS

(D)

PQ RS2 2

2

[ JEE '2001 (Screening) 1 out of 35]

(b) Let 2x2 + y2 – 3xy = 0 be the equation of a pair of tangents drawn from the origin 'O' to a circleof radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

[JEE '2001 (Mains) 5 out of 100]

Q.5 (a) Find the equation of the circle which passes through the points of intersection of circlesx2 + y2 – 2x – 6y + 6 = 0 and x2 + y2 + 2x – 6y + 6 = 0 and intersects the circlex2 + y2 + 4x + 6y + 4 = 0 orthogonally. [REE '2001 (Mains) 3 out of 100]

(b) Tangents TP and TQ are drawn from a point T to the circle x2 + y2 = a2. If the point T lies on theline px + qy = r, find the locus of centre of the circumcircle of triangle TPQ.

[ REE '2001 (Mains) 5 out of 100 ]

Q.6 (a) If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line5x – 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is(A) 4 (B) 2 5 (C) 5 (D) 3 5

(b) If a > 2b > 0 then the positive value of m for which y = mx – b 2m1 is a common tangent tox2 + y2 = b2 and (x – a)2 + y2 = b2 is

(A) 22 b4a

b2

(B)

b2

b4a 22 (C)

b2a

b2

(D)

b2a

b

[ JEE '2002 (Scr)3 + 3 out of 270]

Q.7 The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circlex2 + y2 – 2x – 6y + 6 = 0(A) 1 (B) 2 (C) 3 (D) 3 [JEE '2004 (Scr)]

Q.8 Line 2x + 3y + 1 = 0 is a tangent to a circle at (1, -1). This circle is orthogonal to a circle which is drawnhaving diameter as a line segment with end points (0, –1) and (– 2, 3). Find equation of circle.

[JEE '2004, 4 out of 60]

Q.9 A circle is given by x2 + (y – 1)2 = 1, another circle C touches it externally and also the x-axis, then thelocus of its centre is(A) {(x, y) : x2 = 4y} {(x, y) : y 0} (B) {(x, y) : x2 + (y – 1)2 = 4} {x, y) : y 0}(C) {(x, y) : x2 = y} {(0, y) : y 0} (D) {(x, y) : x2 = 4y} {(0, y) : y 0}

[JEE '2005 (Scr)]

Q.10(a) Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let ADbe perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides,then its radius is(A) 3 (B) 2 (C) 3/2 (D) 1

(b) Tangents are drawn from the point (17, 7) to the circle x2 + y2 = 169.Statement-1: The tangents are mutually perpendicular.becauseStatement-2: The locus of the points from which mutually perpendicular tangents can be drawn to thegiven circle is x2 + y2 = 338.



(A) Statement-1 is true, statement-2 is true; statement-2 is correct explanation for statement-1.(B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true. [JEE 2007, 3+3]

Q.11(a) Consider the two curvesC1 : y

2 = 4x ; C2 : x2 + y2 – 6x + 1 = 0. Then,

(A) C1 and C2 touch each other only at one point(B) C1 and C2 touch each other exactly at two points(C) C1 and C2 intersect (but do not touch) at exactly two points(D) C1 and C2 neither intersect nor touch each other

(b) Consider, L1 : 2x + 3y + p – 3 = 0 ; L2 : 2x + 3y + p + 3 = 0,where p is a real number, and C : x2 + y2 + 6x – 10y + 30 = 0.STATEMENT-1 : If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C.andSTATEMENT-2 : If line L1 is a diameter of circle C, then line L2 is not a chord of circle C.(A) Statement-1 is True, Statement-2 is True; statement-2 is a correct explanation for statement-1(B) Statement-1 is True, Statement-2 is True; statement-2 is NOT a correct explanation for statement-1(C) Statement-1 is True, Statement-2 is False(D) Statement-1 is False, Statement-2 is True

(c) Comprehension (3 questions together):A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides

PQ, QR, RP are D, E, F respectively. The line PQ is given by the equation 3 x + y – 6 = 0 and the

point D is

2

3,

2

33. Further, it is given that the origin and the centre of C are on the same side of the

line PQ.

(i) The equation of circle C is

(A) (x – 32 )2 + (y – 1)2 = 1 (B) (x – 32 )2 + (y + 2

1)2 = 1

(C) (x – 3 )2 + (y + 1)2 = 1 (D) (x – 3 )2 + (y – 1)2 = 1

(ii) Points E and F are given by

(A)

2

3,

2

3, 0,3 (B)

2

1,

2

3, 0,3

(C)

2

3,

2

3,

2

1,

2

3(D)

2

3,

2

3,

2

1,

2

3

(iii) Equations of the sides RP, RQ are

(A) y = 3

2x + 1, y = –

3

2x – 1 (B) y =

3

1x, y = 0

(C) y = 2

3x + 1, y = –

2

3x – 1 (D) y = 3 x, y = 0

[JEE 2008, 3+3 + 4 + 4 + 4]



Q.12 Tangents drawn from the point P(l, 8) to the circle x2 + y2 – 6x – 4y – 11 = 0 [IIT 2009]

touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

(A) x2 + y2 + 4x – 6x + 19 = 0 (B) x2 + y2 – 10y + 19 = 0

(C) x2 + y2 – 2x + 6y – 29 = 0 (D) x2 + y2 – 6x – 4y + 19 = 0

Q. 13 The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let

P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles

C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to

C2 and C, then the radius of the circle C is [JEE 2009, 3 + 4]

Q. 14 The circle passing through the point (–1, 0) and touching the y-axis at (0, 2) also passes through the point

(A) 3

,02

(B) 5

,22

(C)3 5

,2 2

(D) (– 4, 0) [IIT 2011]

Q. 15 The straight line 2x – 3y = 1 divides the circular region x2 + y2 6 into two parts. If

S = 3 5 3 1 1 1 1

2, , , , , , ,4 4 4 4 4 8 4

[IIT 2011]

then the number of point(s) in S lying inside the smaller part is

Q. 16 The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line

4x – 5y = 20 to the circle x2 + y2 = 9 is [IIT 2012]

(A) 20(x2 + y2) – 36x + 45y = 0 (B) 20(x2 + y2) + 36x – 45y = 0

(C) 36(x2 + y2) – 20x + 45y = 0 (D) 36(x2 + y2) + 20x – 45y = 0

Paragraph for Question No. 17 to 18

A tangent PT is drawn to the circle x2 + y2 = 4 at the point P ( 3,1) . A straight line L, perpendicular to

PT is a tangent to the circle (x – 3)2 + y2 = 1 [IIT 2012]

Q. 17 A common tangent of the two circles is

(A) x = 4 (B) y = 2 (C) x + 3 y = – 1 (D) x + 2 2 y = 6

Q. 18 A possible equation of L is

(A) x – 3 y = + 1 (B) x + 3 y = 1 (C) x – 3 y = – 1 (D) x + 3 y = 5

Q. 19 The circle passing through (1, – 2) and touching the axis of x at (3, 0) also passes through the point :

(A) (–2, 5) (B) (–5, 2) (C) (2, –5) (D) (5, –2)

[IIT JEE Main 2013]

Q. 20 Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2 7 on y-axis

is (are) [IIT JEE Adv. 2013](A) x2 + y2 – 6x + 8y + 9 = 0 (B) x2 + y2 – 6x + 7y + 9 = 0

(C) x2 + y2 – 6x – 8y + 9 = 0 (D) x2 + y2 – 6x – 7y + 9 = 0

Q. 21 Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through

origin and touching the circle C externally, then the radius of T is equal to [IIT JEE Main 2014]

(A) 1

4(B)

3

2(C)

3

2(D)

1

2



Q. 22 A circle S passes through the point (0, 1) and is orthogonal to the circles (x – 1)2 + y2 = 16 and

x2 + y2 = 1. Then, [IIT JEE Adv. 2014]

(A) radius of S is 8 (B) radius of S is 7

(C) centre of S is (–7, 1) (D) centre of S is (–8, 1)

Q. 23 The number of common tangents to the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0

is : [IIT JEE Main 2015]

(A) 4 (B) 1 (C) 2 (D) 3

Q. 24 Locus of the image of the point (2, 3) in the line (2x – 3y + 4) + k(x – 2y + 3) = 0, k R, is a

(A) Circle of radius 3 (B) straight line parallel to x-axis

(C) straight line parallel to y-axis (D) circle of radius 2 [IIT JEE Main 2015]

Q. 25 The centres of those circles which touch the circle, x2 – y2 – 8x – 8y – 4 = 0, externally and also touch

the x-axis, lie on : [IIT JEE Main 2016]

(A) a circle (B) an ellipse which is not a circle

(C) a hyperbola (D) a parabola

Q. 26 If one of the diameters of the circle, given by the equation, x2 + y2 – 4x + 6y – 12 = 0, is a chord of a

circle S, whose centre is at (–3, 2), then the radius of S is : [IIT JEE Main 2016]

(A) 5 2 (B) 5 3 (C) 5 (D) 10

Q. 27 Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle,

x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is

(A) x2 + y2 – 4x + 8y + 12 = 0 (B) x2 + y2 – x + 4y – 12 = 0 [IIT JEE Main 2016]

(C) x2 + y2 – x

4+ 2y – 24 = 0 (D) x2 + y2 – 4x + 9y + 18 = 0

Q. 28 The circle C1 : x2 + y2 = 3, with centre at O, intersects the parabola x2 = 2y at the point P in the first

quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3,

respectively. Suppose C2 and C3 have equal radii 2 3 and centres Q2 and Q3, respectively. If Q2 and

Q3 lie on the y-axis, then [IIT JEE Adv. 2016]

(A) Q2Q3 = 12 (B) R2R3 = 4 6

(C) area of the triangle OR2R3 is 6 2 (D) area of the triangle PQ2Q3 is 4 2

Q. 29 Let RS be the diameter of the circle x2 + y2 = 1, where S is the point (1, 0). Let P be a variable point

(other than R and S) on the circle and tangents to the cricle at S and P meet at the point Q. The normal

to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes

through the point(s). [IIT JEE Adv. 2016]

(A) 1 1

,3 3

(B) 1 1

,4 2

(C) 1 1

,3 3

(D)

1 1,

4 2

Q. 30 Let R. Consider the system of linear equations

x + 2y =

3x – 2y =



Which of the following statement(s) is(are) correct ? [IIT JEE Adv. 2016]

(A) If = –3, then the system has infinitely many solutions for all values of and

(B) If –3, then the system has a unique solution for all values of and .

(C) If + = 0, then the system has infinitely many solutions for = –3

(D) If + 0, then the system has no solution for = –3

Q. 31 For how many values of p, the circle x2 + y2 + 2x + 4y – p = 0 and the coordinate axes have exactly three

common points ? [JEE Adv. 2017]

Paragraph "X"Let S be the circle in the xy-plane defined by the equation x2 + y2 = 4. [JEE Adv. 2018]

(There are two question based on Paragraph "X", the question given below is one of them)

Q. 32 Let E1E

2 and F

1F

2 be the chord of S passing through the point P

0(1, 1) and parallel to the x-axis and the

y-axis, respectively. Let G1G

2 be the chord of S passing through P

0 and having slop –1. Let the tangents

to S at E1 and E

2 meet at E

3, the tangents of S at F

1 and F

2 meet at F

3, and the tangents to S at G

1 and

G2 meet at G

3. Then, the points E

3, F

3 and G

3 lie on the curve

(A) x + y = 4 (B) (x – 4)2 + (y – 4)2 = 16(C) (x – 4) (y – 4) = 4 (D) xy = 4

Q. 33 Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect

the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the

curve -

(A) (x + y)2 = 3xy (B) x2/3 + y2/3 = 24/3

(C) x2 + y2 = 2xy (D) x2 + y2 = x2y2

Q. 34 Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let F1 be the set of all pairs of circles

(S1, S2) such that T is tangents to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each

other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1.

Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point

R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the

following statement(s) is (are) TRUE ? [JEE Adv. 2018]

(A) The point (–2, 7) lies in E1 (B) The point 4 7

,5 5

does NOT lie in E2

(C) The point 1

,12

lies in E2 (D) The point 3

0,2

does NOT lie in E1



Q.35 If the circles x2 + y2 + 5kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y–1 = 0, (K R), intersect at thepoints P and Q, then the line 4x + 5y – K = 0 passes through P and Q for : [JEE Main 2019](A) exactly two values of K (B) exactly one value of K(C) no value of K (D) infinitely many values of K

Q.36 The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, –3), thenits radius is : [JEE Main 2019]

(A) 3 2 (B) 3 (C) 2 2 (D) 2

Q.37 A rectangle is inscribed in a circle with a diameter lying along the line, 3y = x + 7. If the two adjacentvertices of the rectangle are (–8, 5) and (6,5), then the area of the rectangle (in sq. units) is:-

[JEE Main 2019]

(A) 72 (B) 84 (C) 98 (D) 56

Q.38 The tangent and the normal lines at the point 3,1 to the circle x2 + y2 = 4 and the x-axis form a

triangle. The area of this triangle (in square units) is : [JEE Main 2019]

(A) 1

3(B)

4

3(C)

1

3(D)

2

3

Q.39 Let C1 and C

2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2–6x– 6y+14 =0

respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the

quadrilateral PC1QC

2 is: [JEE Main 2019]

(A) 8 (B) 6 (C) 9 (D) 4

Q.40 Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point

(0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the

centres of these circles is : [JEE Main 2019]

(A) 1 (B) 2 (C) 2 2 (D) 2

Q.41 If the circles x2 + y2 – 16x – 20 y + 164 = r2 and (x – 4)2 + (y – 7)2 = 36 intersect at two distinct points,

then : [JEE Main 2019]

(A) 0 < r < 1 (B) 1 < r < 11 (C) r > 11 (D) r = 11

Q.42 Let the equations of two sides of a triangle be 3x – 2y + 6 = 0 and 4x + 5y – 20 = 0. If the orthocentre

of this triangle is at (1, 1) then the equation of its third side is :- [JEE-MAIN 2019]

(A) 122 y – 26 x – 1675 = 0 (B) 26 x + 61 y + 1675 = 0

(C) 122 y + 26 x + 1675 = 0 (D) 26 x – 122 y – 1675 = 0

Q.43 Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x - axis as a common

tangent, then: [JEE Main 2019]

(A) 1 1 1

a b c (B) a, b, c are in A.P.

(C) a , b, c are in A.P.. (D) 1 1 1

b a c



Q.44 If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the

point (1, –1), then the radius of C is [JEE Main 2019]

(A) 57 (B) 4 (C) 2 5 (D) 5

Q.45 If the area of an equilateral triangle inscribed in the circle, x2 + y2 + 10x + 12y + c = 0 is 27 3 sq. units

then c is equal to : [JEE Main 2019]

(A) 20 (B) 25 (C) 13 (D) –25

Q.46 A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinateaxes. Then the distance of the vertex of this square which is nearest to the origin is : [JEE Main 2019]

(A) 13 (B) 137 (C) 6 (D) 41

Q.47 A circle cuts a chord of length 4a on the x - axis and passes through a point on the y - axis, distance 2b

from the origin. Then the locus of the centre of this circle, is :-

[JEE Main 2019]

(A) A hyperbola (B) A parabola (C) A straight line (D) An ellipse

Q.48 If a variable line, 3x + 4y – = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and

x2 + y2 – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of is the interval:-

[JEE Main 2019]

(A) [12, 21] (B) (2, 17) (C) (23, 31) (D) [13, 23]

Q.49 The sum of squares of lengths of the chords intercepted on the circle x2 + y2 = 16, by the lines,

x + y = n, n N , where N is the set of all natural numbers, is: [JEE Main 2019](A) 320 (B) 160 (C) 105 (D) 210

Q.50 If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then thelocus of the mid - point of PQ is :

[JEE Main 2019](A) x2 + y2 – 2xy = 0 (B) x2 + y2 – 16x2y2 = 0(C)x2 + y2 – 4x2y2 = 0 (D) x2 + y2 – 2x2y2 = 0

Q.51 The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y – 24 =0 also passes through thepoint:- [JEE Main 2019](A) (–4, 6) (B) (6, –2) (C) (–6, 4) (D) (4, –2)

Q.52 The tangent to the curve, 2xy xe passing, through the point (1, e) also passes through the point :

[JEE Main 2019]

(A) (2, 3e) (B) 4

,2e3

(C) 5

, 2e3

(D) (3, 6e)

Q.53 A line y = mx + 1 intersects the circle (x – 3)2 + (y + 2)2 = 25 at the points P and Q. If the midpoint of

the line segment PQ has x-coordinate 3

5 , then which one of the following options is correct ?

[JEE Advanced 2019](A) 6 m < 8 (B) 2 m < 4 (C) 4 m < 6 (D) – 3 m < – 1



Q.54 Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let A and

B be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles

A and

B such that both the circles are on the same side of T. If C is the point of intersection of T and

the line passing through A and B, then the length of the line segment AC is _______[JEE Advanced 2019]

Q.55 Let the tangents drawn from the origin to the circle, x2 + y2 – 8x – 4y + 16 = 0 touch it at the points Aand B. The (AB)2 is equal to : [JEE Main 2020]

(A) 32

5(B)

64

5(C)

52

5(D)

56

5

Q.56 If a line, y = mx + c is a tangent to the circle (x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where

L1 is the tangent to the circle, x2 + y2 = 1 at the point

1 1,

2 2

; then : [JEE Main 2020]

(A) c2 + 7c + 6 = 0 (B) c2 – 6c + 7 = 0 (C) c2 – 7c + 6 = 0 (D) c2 + 6c + 7 = 0

Q.57 A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the followinglines is not a tangent to this circle ? [JEE Main 2020](A) 4x – 3y + 17 = 0 (B) 3x + 4y – 6 = 0 (C) 4x + 3y – 8 = 0 (D) 3x – 4y – 24 = 0

Q.58 The number of integral values of k for which the line, 3x + 4y = k intersects the circle,x2 + y2 –2x – 4y + 4 = 0 at two distinct points is................ [JEE Main 2020]

Q.59 The set of all possible values of in the interval (0, ) for which the points (1,2) and (sin , cos ) lie onthe same side of the line x + y = 1 is: [JEE Main 2020]

(A) 0,4

(B) 0,2

(C) 3

0,4

(D) 3

,4 4

Q.60 The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touchesboth the lines x = 3 and y = 2, is .......... [JEE Main 2020]

Q.61 The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having itscentre on the line, 2x – 3y + 12 = 0, also passes through the point: [JEE Main 2020](A) (–1, 3) (B) (1, –3) (C) (–3,6) (D) (–3,1)

Q.62 If the curves, x2 – 6x + y2 + 8 = 0 and x2 – 8x + y2 + 16 – k = 0, (k > 0) touch each other at a point, thenthe largest value of k is _______ [JEE Main 2020]

Q.63 If the length of the chord of the circle, x2 + y2 = r2 (r > 0) along the line, y – 2x = 3 is r, then r2 is equal to:[JEE Main 2020]

(A) 9/5 (B) 12 (C) 24/5 (D) 12/5

Q.64 Let PQ be diameter of the circle x2 + y2 = 9. If and are the lengths of the perpendiculars from P andQ on the straight line, x + y = 2 respectively, then the maximum value of is ____.

[JEE Main 2020]

Q.65 Let O be the centre of the circle x2 + y2 = r2, where 5

r2

. Suppose PQ is a chord of this circle and

the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of thetriangle OPQ lies on the line x + 2y = 4, then the value of r is [JEE Advanced 2020]



ANSWER KEYELEMENTARY EXERCISES


Q.1 x2 + y2 + 2x – 4y = 4 Q.2 x2 + y2 + 10x + 12y = 39

Q.3 x2 + y2 – 2ax + 2by = 2ab Q.4 x2 + y2 + 2ax + 2by + 2b2 = 0

Q.5 (2, 4); 61 Q.65

,16

; 1

136

Q.7k

,02

; 5 k

2Q.8 (g, –f); 2 2f g

Q.92 2

c mc,

1 m 1 m

; |c| Q.13 15x2 + 15y2 – 94x + 18y + 55 = 0

Q.14 b(x2 + y2 – a2) = x(b2 + h2 – a2) Q.15 x2 + y2 – ax – by = 0

Q.16 x2 + y2 – 22x – 4y + 25 = 0 Q.17 x2 + y2 – 5x – y + 4 = 0

Q.18 3x2 + 3y2 – 29x – 19y + 56 = 0 Q.19 b(x2 + y2) – (a2 + b2)x + (a – b) (a2 + b2) = 0

Q.21 x2 + y2 – 3x – 4y = 0 Q.222 2

2 2 a bx y (x y) 0

a b

;

2 2a b

a b

Q.23 x2 + y2 – hx – ky = 0 Q.242 2 2 2 2x y 2 a b y – b 0

Q.25 x2 + y2 – 10x – 10y + 25 = 0 Q.26 x2 + y2 – 2ax – 2ay + a2 = 0

Q.27 2 2x y 2(5 12)(x y) 37 10 12 0 Q.28 x2 + y2 – 6x + 4y + 9 = 0

Q.29 b(x2 + y2) = x(b2 + c2) Q.30 2 2x y 6 2y 6x 9 0

Q.31 x2 + y2 – 3x + 2 = 0; 2 22x 2y 5x 3y 3 0 ; 2 22x 2y 7x 3y 6 0

Q.33 (x + 21)2 + (y + 13)2 = 652 Q.34 8x2 + 8y2 – 25x – 3y + 18 = 0

Q.36 x2 + y2 = a2 + b2; x2 + y2 – 2(a + b) x + 2 (a – b)y + a2 + b2 = 0


Q.1 5x – 12y = 152 Q.2 24x + 10y + 151 = 0

Q.3 x 2y 2 5 Q.4 2 2x 2y g 2f 5 g f c

Q.5c c

,2 2

Q.6 c = a; (0, b)

Q.7 Yes Q.8 k = 40 or – 10

Q.9 2 2 2 2 2a cos bsin a b sin Q.10 2 2Aa Bb C c A B



Q.11 (1) 2y mx a 1 m ; (2) 2my x a 1 m ; (3) 2 2ax y b a ab ; (4) x y a 2

Q.122 2

2

2 2

a b2 r

a b

Q.13 2 2x y 2ax 0 ; 2 2x y 2ay 0

Q.14 c = b – am; 2 2 2c b am (1 m )(a b ) Q.15 2 2 381x y 6x 8y 0

169

Q.16 x2 + y2 – 2cx – 2cy + c2 = 0, where 2 22c a b a b

Q.17 5x2 + 5y2 – 10x + 30y + 49 = 0 Q.18 x2 + y2 – 2cx – 2cy + c2 = 0

Q.19 (x – r)2 + (y – h)2 = r2 Q.20 x2 + y2 – 2x – 2y = 0


Q.1 x + 2y = 7 Q.2 8x – 2y = 11

Q.3 x = 0 Q.4 23x + 5y = 57

Q.5 by – ax = a2 Q.6 (5, 10)

Q.73 3

,5 10

Q.8 (1, – 2)

Q.91 1

,2 3

Q.10 (–2a, –2b)

Q.1118

6,5

Q.12 3y – 2x = 13; 162 243

,13 13

Q.13 (2, –1) Q.14 2 2 2x y 2a

Q.181

462

Q.19 9

Q.20 2 22a 2ab b Q.2133 1

,2 ;4 4

Q.23 (1) 28x2 + 33xy – 28y2 – 715x – 195y + 4225 = 0

(2) 123x2 – 64xy + 3y2 – 664x + 226y + 763 = 0

EXERCISE–IQ.1 square of side 2; x2 + y2 = 1; x2 + y2 = 2

Q.2 zero, zero

Q.4 32 sq. unit

Q.5 x y = 0 ; x + 7y = 0

Q.6 (5, 1) & (–1, 5)

Q.7 4x 3y 25 = 0 OR 3x + 4y 25 = 0



Q.8 (i) (11, 16), (ii) (11, 8), (iii) (11, 12)

Q.9 x2 + y2 – 2x – 2y + 1 = 0 OR x2 + y2 – 42x + 38y – 39 = 0

Q.10 (i) 3x – 4y = 21; 4x + 3y = 3; (ii) A(0, 1) and B (–1, – 6); (iii) 90°, 125 units

(iv) 25 sq. units, 12.5 sq. units; (v) x2 + y2 + x + 5y – 6, x intercept 5; y intercept 7

Q.11 x2 + y2 – 2x – 2y = 0

Q.12 2x – 2y – 3 = 0

Q.13 a2(x2 + y2) = 4x2y2

Q.14 x2 + y2 = a2 + b2; r = 22 ba

Q.15 2(x2 + y2) + 6x – 17y – 6 = 0

Q.16 63

Q.17 6

Q.18 x 7y = 2, 7x + y = 14; (x 1)2 + (y 7)2 = 32; (x 3)2 + (y + 7)2 = 32 ;

(x 9)2 + (y 1)2 = 32; (x + 5)2 + (y + 1)2 = 32

Q.20 x + y = 2

Q.21 10

Q.22 r = 15

Q.23 19

Q.24 (– 4, 2), x2 + y2 – 2x – 6y – 15 = 0

Q.25 x2 + y2 – 6x + 4y=0 OR x2 + y2 + 2x – 8y + 4=0

Q.26 x2 + y2 + x 6y + 3 = 0

Q.27 5x2 + 5y2 – 8x – 14y – 32 = 0

Q.29 64

Q.30 3

EXERCISE–II

Q.2 75 sq. unit Q.3 19

Q.5 4 Q.7 x2 + y2 + 16x + 14y – 12 = 0

Q.8 ( 4, 4) ; (– 1/2, 1/2) Q.9 (a) x2 + y2 + 4x – 6y = 0; k = 1; (b) x2 + y2 = 64

Q.10 9x 10y + 7 = 0; radical axis Q.11 x2 + y2 + 7x 11y + 38 = 0

Q.12 x2 + y2 + 6x – 3y = 0 Q.13

3

23,2

Q.14 x2 + y2 3x 3y + 4 = 0 Q.15 (1, 0) & (1/2,1/2); r = 22

1

Q.16 4x2 + 4y2 + 6x + 10y – 1 = 0 Q.17 40

Q.18 x2 + y2 – 12x – 12y + 64 = 0 Q.19 x2 + y2 ± a 2 x = 0

Q.20 (A) S; (B) R ; (C) Q ; (D) P



EXERCISE–III

Q. 1 (a) D (b) B, C (c) c1 : (x – 4)2 + y2 = 9 c2 : 2

4x

3

+ y2 =

1

9

common tangent between c & c1 : T1 = 0 ; T2 = 0 and x – 1 = 0 ;

common tangent between c & c2 : T1 = 0 ; T2 = 0 and x + 1 = 0

common tangent between c1 & c2 : T1 = 0 ; T2 = 0 and y = ± 4

x5

5

39

where T1 : x – 3 y + 2 = 0 and T2 : x + 3 y + 2 = 0

Q.2 (a) C (b) A

Q.3 (a) 6 x 8 y + 25 = 0 & 6 x 8 y 25 = 0; (b) (–9/2 , 2)

(c) x2 + y2 + 4x – 12 = 0, T1: 0432yx3 , TT2: 0432yx3 (D.C.T.)

T3: 02y3x , TT4: 06y3x (T.C.T.)

Q.4 (a) A; (b) OA = 3(3 + 10 ) Q.5 (a) x2 + y2 + 14x – 6y + 6 = 0; (b) 2px + 2qy = r

Q.6 (a) C; (b) A Q.7 C

Q.8 2x2 + 2y2 – 10x – 5y + 1 = 0 Q.9 D Q.10 (a) B; (b) A

Q.11 (a) B; (b) C; (c) (i) D, (ii) A, (iii) D Q.12 B;

Q. 13 8 Q. 14 D Q. 15 2 Q. 16 A Q. 17 D

Q. 18 A Q. 19 D Q. 20 A, C Q. 21 A Q. 22 B, C

Q. 23 D Q. 24 D Q. 25 D Q. 26 B Q. 27 A

Q. 28 A, B, C Q. 29 A, C Q. 30 B, C, D Q. 31 2 Q. 32 A

Q. 33 D Q. 34 B, D Q.35 C Q.36 C Q.37 B

Q.38 D Q.39 D Q.40 B Q.41 B Q.42 D

Q.43 A Q.44 D Q.45 B Q.46 D Q.47 B

Q.48 A Q.49 D Q.50 C Q.51 B Q.52 B

Q.53 B Q.54 10.00 Q.55 B Q.56 D Q.57 C

Q.58 9 Q.59 B Q.60 3 Q.61 C Q.62 4

Q.63 D Q.64 7 Q.65 2



QUESTION BANK

Single Correct Type :

Q.1 Coordinates of the centre of the circle which bisects the circumferences of the circlesx2 + y2 = 1 ; x2 + y2 + 2x – 3 = 0 and x2 + y2 + 2y – 3 = 0 is

(A) (–1, 1) (B) (3, 3) (C) (2, 2) (D) (–2, –2)

*Q.2 The ends of a quadrant of a circle have the coordinates (1, 3) and (3, 1) then the centre of the such acircle is(A) (1, 1) (B) (2, 2) (C) (2, 6) (D) (4, 4)

* Q.3 Let C be a circle x2 + y2 = 1. The line l intersect C at the point (–1, 0) and the point P. Suppose that theslope of the line l is a rational number m. Number of choices for m for which both the coordinates of Pare rational, is(A) 3 (B) 4 (C) 5 (D) infinitely many

* Q.4 The line 2x – y + 1 = 0 is tangent to the circle at the point (2, 5) and the centre of the circles lies onx – 2y = 4. The radius of the circle is

(A) 53 (B) 35 (C) 52 (D) 25

Q.5 One circle has a radius of 5 and its center at (0, 5). A second circle has a radius of 12 and its centre at(12, 0). The length of a radius of a third circle which passes through the centre of the second circle andboth points of intersection of the first 2 circles, is equal to(A) 13/2 (B) 15/2 (C) 17/2 (D) none

* Q.6 Consider 3 non collinear points A, B, C with coordinates (0, 6), (5, 5) and (–1, 1) respectively. Equationof a line tangent to the circle circumscribing the triangle ABC and passing through the origin is(A) 2x – 3y = 0 (B) 3x + 2y = 0(C) 3x – 2y = 0 (D) 2x + 3y = 0

Q.7 A circle is inscribed in an equilateral triangle with side lengths 6 unit. Another circle is drawn inside thetriangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. Theradius of the smaller circle is

(A) 1/ 3 (B) 2/3 (C) 1/2 (D) 1

Q.8 To which of the following circles, the line y x + 3 = 0 is normal at the point

2

3,

2

33 ?

(A) 92

3y

2

33x

22

(B) 9

2

3y

2

3x

22

(C) x2 + (y 3)2 = 9 (D) (x 3)2 + y2 = 9

Q.9 The circle with equation x2 + y2 = 1 intersects the line y = 7x + 5 at two distinct points A and B. Let Cbe the point at which the positive x-axis intersects the circle. The angle ACB is

(A) tan–1

3

4(B) tan–1

4

3(C) tan–1(1) (D) tan–1

2

3



*Q.10 A square OABC is formed by line pairs xy = 0 and xy + 1 = x + y where ‘O’ is the origin. A circle withcentre C1 inside the square is drawn to touch the line pair xy = 0 and another circle with centre C2 andradius twice that of C1, is drawn to touch the circle C1 and the other line pair. The radius of the circle withcentre C1 is

(A)

2

3 2 1(B)

2 2

3 2 1(C)

2

3 2 1(D)

2 1

3 2

*Q.11 If the circlesx2 + y2 + 2ax + 2by + c = 0

and x2 + y2 + 2bx + 2ay + c = 0

where c > 0, have exactly one point in common then the value of 2(a b)

2c

is

(A) 1 (B) 2 (C) 2 (D) 1/2

*Q.12 The shortest distance from the line 3x + 4y = 25 to the circle x2 + y2 = 6x – 8y is equal to(A) 7/5 (B) 9/5 (C) 11/5 (D) 32/5

*Q.13 Two circles with centres at A and B, touch at T. BD is the tangent at D and TC is a common tangent.AT has length 3 and BT has length 2. The length CD is(A) 4/3 (B) 3/2 (C) 5/3 (D) 7/4

B AT

CD

Q.14 Triangle ABC is right angled at A. The circle with centre A and radius AB cuts BC and AC internally atD and E respectively. If BD = 20 and DC = 16 then the length AC equals

(A) 216 (B) 266 (C) 30 (D) 32

Q.15 From the point A(0, 3) on the circle x2 + 4x + (y – 3)2 = 0 a chord AB is drawn & extended to a pointM such that AM = 2 AB. The equation of the locus of M is :(A) x2 + 8x + y2 = 0 (B) x2 + 8x + (y – 3)2 = 0(C) (x – 3)2 + 8x + y2 = 0 (D) x2 + 8x + 8y2 = 0

Q.16 If x = 3 is the chord of contact of the circle x2 y2 = 81, then the equation of the corresponding pair oftangents, is(A) x2 8y2 + 54x + 729 = 0 (B) x2 8y2 54x + 729 = 0(C) x2 8y2 54x 729 = 0 (D) x2 8y2 = 729

Q.17 From (3, 4) chords are drawn to the circle x2 + y2 – 4x = 0. The locus of the mid points of the chords is(A) x2 + y2 – 5x – 4y + 6 = 0 (B) x2 + y2 + 5x – 4y + 6 = 0(C) x2 + y2 – 5x + 4y + 6 = 0 (D) x2 + y2 – 5x – 4y – 6 = 0

*Q.18 The centre of the smallest circle touching the circles x2 + y2 – 2y 3 = 0 andx2 + y2 8x 18y + 93 = 0 is(A) (3 , 2) (B) (4 , 4) (C) (2 , 7) (D) (2 , 5)



Q.19 If the circle C1 : x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the

common chord is of maximum length and has a slope equal to 3/4, then the co-ordinates of the centre ofC2 are :

(A)

9

5

12

5, (B)

9

5

12

5, (C)

12

5

9

5, (D)

12

5

9

5,

Q.20 In the diagram, DC is a diameter of the large circle centered at A, and AC is a diameter of the smallercircle centered at B. If DE is tangent to the smaller circle at F and DC = 12 then the length of DE is

(A) 8 2 (B) 16 (C) 9 2 (D) 10 2

D C

EF

A B

Q.21 Let C be the circle of radius unity centred at the origin. If two positive numbers x1 and x2 are such thatthe line passing through (x1, – 1) and (x2, 1) is tangent to C then(A) x1x2 = 1 (B) x1x2 = – 1 (C) x1 + x2 = 1 (D) 4x1x2 = 1

Q.22 The locus of the centers of the circles which cut the circles x2 + y2 + 4x 6y + 9 = 0 andx2 + y2 5x + 4y 2 = 0 orthogonally is :(A) 9x + 10y 7 = 0 (B) x y + 2 = 0(C) 9x 10y + 11 = 0 (D) 9x + 10y + 7 = 0

Q.23 The locus of the center of the circles such that the point (2, 3) is the mid point of the chord 5x + 2y = 16is(A) 2x – 5y + 11 = 0 (B) 2x + 5y – 11 = 0 (C) 2x + 5y + 11 = 0 (D) none

Q.24 The locus of the mid points of the chords of the circle x² + y² + 4x 6y 12 = 0 which subtend an angle

of

3 radians at its circumference is :

(A) (x 2)² + (y + 3)² = 6.25 (B) (x + 2)² + (y 3)² = 6.25(C) (x + 2)² + (y 3)² = 18.75 (D) (x + 2)² + (y + 3)² = 18.75

Q.25 In a circle with centre ‘O’ PA and PB are two chords. PC is the chord that bisects the angle APB. Thetangent to the circle at C is drawn meeting PA and PB extended at Q and R respectively. If QC = 3,QA = 2 and RC = 4, then length of RB equals(A) 2 (B) 8/3 (C) 10/3 (D) 11/3

*Q.26 Suppose that two circles C1 and C2 in a plane have no points in common. Then(A) there is no line tangent to both C1 and C2(B) there are exactly four lines tangent to both C1 and C2(C) there are no lines tangent to both C1 and C2 or there are exactly two lines tangent to both C1 and C2(D) there are no lines tangent to both C1 and C2 or there are exactly four lines tangent to both C1 and C2

Q.27 If two chords of the circle x2 + y2 – ax – by = 0, drawn from the point (a, b) is divided by the x-axis inthe ratio 2 : 1 then :(A) a2 > 3b2 (B) a2 < 3b2 (C) a2 > 4b2 (D) a2 < 4b2



Q.28 Consider the points P(2, 1) ; Q(0, 0) ; R(4, –3) and the circle S : x2 + y2 – 5x + 2y – 5 = 0(A) exactly one point lies outside S (B) exactly two points lie outside S(C) all the three points lie outside S (D) none of the point lies outside S

Q.29 The angle at which the circles (x – 1)2 + y2 = 10 and x2 + (y – 2)2 = 5 intersect is

(A) 6

(B)

4

(C)

3

(D)

2

Q.30 P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both theco-ordinate axes cut at right angles, then(A) a2 – 6ab + b2 = 0 (B) a2 + 2ab – b2 = 0(C) a2 – 4ab + b2 = 0 (D) a2 – 8ab + b2 = 0

Q.31 Three concentric circles of which the biggest is x2 + y2 = 1, have their radii in A.P. If the line y = x + 1 cutsall the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is

(A) 1

0,4

(B) 1

0,2 2

(C) 2 2

0,4

(D) none

Q.32 A circle is inscribed into a rhombus ABCD with one angle 60°. The distance from the center of the circleto the nearest vertex is equal to 1. If P is any point of the circle, then|PA|2 + |PB|2 + |PC|2 + |PD|2 is equal to(A) 12 (B) 11 (C) 9 (D) none

*Q.33 The value of 'c' for which the set, {(x, y)x2 + y2 + 2x 1} {(x, y)x y + c 0} contains onlyone point in common is :(A) (, 1] [3, )(B) {1, 3} (C) {3} (D) { 1 }

Q.34 A tangent at a point on the circle x2 + y2 = a2 intersects a concentric circle C at two points P and Q. Thetangents to the circle C at P and Q meet at a point on the circle x2 + y2 = b2 then the equation of circle‘C’ is(A) x2 + y2 = ab (B) x2 + y2 = (a – b)2

(C) x2 + y2 = (a + b)2 (D) x2 + y2 = a2 + b2

Q.35 Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles,x2 + y2 – ( + 6)x + (8 – 2)y – 3 = 0 . being the variable. The locus of the point of intersection ofthese tangents is :(A) 2x – y + 10 = 0 (B) x + 2y – 10 = 0 (C) x – 2y + 10 = 0 (D) 2x + y – 10 = 0

Q.36 If aa

,1

, b

b,

1

, c

c,

1

and d

d,

1

are four distinct points on a circle of radius 4 units then,

abcd is equal to(A) 4 (B) 1/4 (C) 1 (D) 16

Q.37 ABCD is a square of unit area. A circle is tangent to two sides of ABCD and passes through exactly oneof its vertices. The radius of the circle is

(A) 2 2 (B) 2 1 (C) 1

2(D)

1

2



Q.38 A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at Aenclosing an angle of 60°. The area enclosed by these tangents and the arc of the circle is

(A) 2

63

(B) 3

3

(C)

3

3 6

(D) 3 1

6

Q.39 A straight line with slope 2 and y-intersect 5 touches the circle, x2 + y2 + 16x + 12y + c = 0 at a point Q.Then the coordinates of Q are(A) (–6, 11) (B) (–9, –13) (C) (–10, –15) (D) (–6, –7)

Q.40 A foot of the normal from the point (4, 3) to a circle is (2, 1) and a diameter of the circle has the equation2x – y – 2 = 0. Then the equation of the circle is(A) x2 + y2 – 4y + 2 = 0 (B) x2 + y2 – 4y + 1 = 0(C) x2 + y2 – 2x – 1 = 0 (D) x2 + y2 – 2x + 1 = 0

Q.41 AB is a diameter of a circle. CD is a chord parallel to AB and 2 CD = AB. The tangent at B meets theline AC produced at E then AE is equal to :

(A) AB (B) 2 AB (C) 2 2 AB (D) 2 AB

Q.42 A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of coordinates in points Pand Q, then the equation of the locus of the foot of perpendicular from O to PQ is :

(A) (x2 + y2) 1 12 2x y

= 4 a

2 (B) (x2 + y2)2 1 12 2x y

= a2

(C) (x2 + y2)2 1 12 2x y

= 4

a2 (D) (x2 + y2) 1 12 2x y

= a2

Q.43 If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and Bthen the locus of the centroid of the triangle OAB is(A) x2 + y2 = (2k)2 (B) x2 + y2 = (3k)2 (C) x2 + y2 = (4k)2 (D) x2 + y2 = (6k)2

Q.44 Tangents are drawn from (4, 4) to the circle x2 + y2 – 2x – 2y – 7 = 0 to meet the circle at A and B. Thelength of the chord AB is

(A) 2 3 (B) 3 2 (C) 2 6 (D) 6 2

Q.45 Points P and Q are 3 units apart. A circle centre at P with a radius of 3 units intersects a circle centred at

Q with a radius of 3 units at point A and B. The area of the quadrilateral APBQ is

(A) 99 (B) 99

2(C)

99

2(D)

99

16

Q.46 Tangents are drawn from any point on the circle x2 + y2 = R2 to the circle x2 + y2 = r2. If the line joiningthe points of intersection of these tangents with the first circle also touch the second, then R equals

(A) 2 r (B) 2r (C) 2r

2 3(D)

4r

3 5

Q.47 The equation of the circle symmetric to the circle x2 + y2 – 2x – 4y + 4 = 0 about the line x – y = 3 is(A) x2 + y2 – 10x + 4y + 28 = 0 (B) x2 + y2 + 6x + 8 = 0(C) x2 + y2 – 14x – 2y + 49 = 0 (D) x2 + y2 + 8x + 2y + 16 = 0



Q.48 The locus of the centre of a circle which touches externally the circle, x2 + y2 – 6x – 6y + 14 = 0 & alsotouches the y-axis is given by the equation :(A) x2 – 6x – 10y + 14 = 0 (B) x2 – 10x – 6y + 14 = 0(C) y2 – 6x – 10y + 14 = 0 (D) y2 – 10x – 6y + 14 = 0

Q.49 The equation of the locus of the mid points of the chords of the circle 4x2 + 4y2 – 12x + 4y + 1 = 0 that

subtend an angle of 2

3

at its centre is

(A) 16(x2 + y2) – 48x + 16y + 31 = 0 (B) 16(x2 + y2) – 48x – 16y + 31 = 0(C) 16(x2 + y2) + 48x + 16y + 31 = 0 (D) 16(x2 + y2) + 48x – 16y + 31 = 0

Q.50 Number of value(s) of A for which the system of equations x2 = y2 and (x – A)2 + y2 = 1 has exactly 3solutions, is(A) 1 (B) 2 (C) 3 (D) 4

Q.51 A variable circle C has the equationx2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter.

If the power of point P(a,b) w.r.t. the circle C is constant then the ordered pair (a, b) is

(A)

10

1,

10

1(B)

10

1,

10

1(C)

10

1,

10

1(D)

10

1,

10

1

Q.52 Consider the circles, x2 + y2 = 25 and x2 + y2 = 9. From the point A(0, 5) two segments are drawntouching the inner circle at the points B and C while intersecting the outer circle at the points D and E. If‘O’ is the centre of both the circles then the length of the segment OF that is perpendicular to DE, is(A) 7/5 (B) 7/2 (C) 5/2 (D) 3

Paragraph for Question Nos. 53 to 55

Let C be a circle of radius r with centre at O. Let P be a point outside C and D be a point on C. A linethrough P intersects C at Q and R, S is the midpoint of QR.

Q.53 For different choices of line through P, the curve on which S lies, is(A) a straight line (B) an arc of circle with P as centre(C) an arc of circle with PS as diameter (D) an arc of circle with OP as diameter

Q.54 Let P is situated at a distance 'd' from centre O, then which of the following does not equal the product(PQ) (PR)?(A) d2 – r2 (B) PT2, where T is a point on C and PT is tangent to C(C) (PS)2 – (QS)(RS) (D) (PS)2

Q.55 Let XYZ be an equilateral triangle inscribed in C. If , , denote the distances of D from vertices X,Y, Z respectively, the value of product ( + – ) ( + – ) ( + – ), is

(A) 0 (B) 8

(C) 6

3333 (D) None of these



Paragraph for Question Nos. 56 to 58

Consider the circle S : x2 + y2 – 4x – 1 = 0 and the line L : y = 3x – 1. If the line L cuts the circle at A andB then

Q.56 Length of the chord AB equal

(A) 2 5 (B) 5 (C) 5 2 (D) 10

Q.57 The angle subtended by the chord AB in the minor arc of S is

(A) 3

4

(B)

5

6

(C)

2

3

(D)

4

Q.58 Acute angle between the line L and the circle S is

(A) 2

(B)

3

(C)

4

(D)

6

Assertion & Reason

Q.59 Let A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of a triangle ABC.Statement-1 : If angle C is obtuse then the quantity (x3 – x1)(x3 – x2) + (y3 – y1)(y3 – y2) is negative.becauseStatement-2 : Diameter of a circle subtends obtuse angle at any point lying inside the semicircle.(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.(B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.(C) Statement-1 is true, Statement-2 is false(D) Statement-1 is false, Statement-2 is true

Q.60 Statement-1 : Angle between the tangents drawn from the point P(13, 6) to the circleS : x2 + y2 – 6x + 8y – 75 = 0 is 90°

becauseStatement-2 : Point P lies on the director circle of S.(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.(B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.(C) Statement-1 is true, Statement-2 is false(D) Statement-1 is false, Statement-2 is true

Q.61 Consider the circle C : x2 + y2 – 2x – 2y – 23 = 0 and a point P(3, 4).Statement-1 : No normal can be drawn to the circle C, passing through (3, 4).becauseStatement-2 : Point P lies inside the given circle, C.(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.(B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.(C) Statement-1 is true, Statement-2 is false(D) Statement-1 is false, Statement-2 is true

Q.62 Consider the linesL : (k + 7)x – (k – 1)y – 4(k – 5) = 0 where k is a parameter

and the circleC : x2 + y2 + 4x + 12y – 60 = 0

Statement-1 : Every member of L intersects the circle ‘C’ at an angle 90°becauseStatement-2 : Every member of L is tangent to the circle C.(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.(B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.(C) Statement-1 is true, Statement-2 is false(D) Statement-1 is false, Statement-2 is true



One or more than one correct :

Q.63 The equation of a circle with centre (4, 3) and touching the circle x2 + y2 = 1 is :(A) x2 + y2 – 8x – 6y – 9 = 0 (B) x2 + y2 – 8x – 6y + 11 = 0(C) x2 + y2 – 8x – 6y – 11 = 0 (D) x2 + y2 – 8x – 6y + 9 = 0

Q.64 A circle passes through the points (–1, 1), (0, 6) and (5, 5). The point(s) on this circle, the tangent(s) atwhich is/are parallel to the straight line joining the origin to its centre is/are :(A) (1, –5) (B) (5, 1) (C) (–5, –1) (D) (–1, 5)

*Q.65 Point M moved along the circle (x – 4)2 + (y – 8)2 = 20. Then it broke away from it and moving alonga tangent to the circle, cuts the x-axis at the point (–2, 0). The co-ordinates of the point on the circle atwhich the moving point broke away can be :

(A) 3 46

,5 5

(B) 2 44

,5 5

(C) (6, 4) (D) (3, 5)

Q.66 The centre(s) of the circle(s) passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9is/are

(A) 3 1

,2 2

(B) 1 3

,2 2

(C) 1/21

,22

(D) 1/21

, 22

Q.67 Which of the following lines have the intercepts of equal lengths on the circle, x2 + y2 – 2x + 4y = 0 ?(A) 3x – y = 0 (B) x + 3y = 0 (C) x + 3y + 10 = 0 (D) 3x – y – 10 = 0

*Q.68 Consider the circles C1 : x2 + y2 = 16 and C2 : x

2 + y2 – 12x + 32 = 0. Which of the following statementsis/are correct ?(A) Number of common tangent to these circles is 3(B) The point P with coordinates (4, 1) lies outside the circle C1 and inside the C2(C) Their direct common tangent intersect at (12, 0)(D) Slope of their radical axis is not defined

Q.69 Consider the circlesC1 : x

2 + y2 – 4x + 6y + 8 = 0C2 : x

2 + y2 – 10x – 6y + 14 = 0Which of the following statement(s) hold good in respect of C1 and C2 ?(A) C1 and C2 are orthogonal(B) C1 and C2 touch each other(C) Radical axis between C1 and C2 is also one of their common tangent(D) Middle point of the line joining the centres of C1 and C2 lies on their radical axis

*Q.70 Three distinct lines are drawn in a plane. Suppose there exist exactly n circles in the plane tangent to allthe three lines, then the possible values of n is/are(A) 0 (B) 1 (C) 2 (D) 4



ANSWER KEYSingle Correct :

Q.1 D Q.2 A Q.3 D Q.4 A Q.5 A Q.6 D Q.7 A

Q.8 D Q.9 C Q.10 C Q.11 A Q.12 A Q.13 B Q.14 B

Q.15 B Q.16 B Q.17 A Q.18 D Q.19 B Q.20 A Q.21 A

Q.22 C Q.23 A Q.24 B Q.25 B Q.26 D Q.27 A Q.28 D

Q.29 B Q.30 C Q.31 C Q.32 B Q.33 D Q.34 A Q.35 A

Q.36 C Q.37 A Q.38 B Q.39 D Q.40 C Q.41 D Q.42 C

Q.43 A Q.44 B Q.45 B Q.46 B Q.47 A Q.48 D Q.49 A

Q.50 B Q.51 B Q.52 A Q.53 D Q.54 D Q.55 A Q.56 D

Q.57 A Q.58 C Q.59 A Q.60 A Q.61 D Q.62 C

One or more than one correct

Q.63 CD Q.64 BD Q.65 BC Q.66 CD Q.67 ABCD Q.68 ACD

Q.69 BC Q.70 ACD

Circle - Best Approach

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