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Section 3.2 - Circles

Conic Sections:

Circles

The equation of a circle with center ),( kh and radius r is

222 )()( rkyhx

Sketch the graph of the circle or semicircle

Important measurements/ points of a circle?

#24 2 2 7x y

Exercise: (𝑥 + 4)2 + (𝑦 − 2)2 = 25

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Find an equation of the circle that satisfies the stated conditions

#36 Center 4,1 , radius 5C

#40 Center at the origin, passing through 4, 7P

#46 Find the equation of the circle that satisfies the following conditions:

Endpoints of a diameter 𝐴 (−5, 2) and 𝐵 (3, 6)

The technique of completing the square:

Exercise: Solve by completing the square:

𝒙𝟐 + 𝟔𝒙 − 𝟏𝟔 = 𝟎

3

By completing the square for x and y, find the equation of the circle:

#48 2 2 8 10 37 0x y x y

#52 2 29 9 12 6 4 0x y x y

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Section 11.1 – Parabolas (p.734)

Definition of a Parabola

A parabola is the set of all points in a plane equidistant from a fixed point F (the focus) and a fixed line l (the

directrix) that lie in the plane, with F not on l.

If 𝑝 < 0, the parabola opens downwards

If 𝑝 < 0, the parabola opens to the left.

The opening width at the focus is 4p,

it is called the latus rectum

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Find the vertex, focus, and directrix of the following parabolas. Sketch the graph, showing the focus and the

directrix.

#4 220x y

#6 2 1

3 12

x y

#11 𝑦2 + 14𝑦 + 4𝑥 + 45 = 0

6

Find an equation of the parabola that satisfies the given conditions

#24 Focus 𝐹(−3, −2), directrix 𝑦 = 1

#26 Vertex 2,3V , directrix 𝑥 = 1

#32 Vertex (4, 7) , focus 𝐹(4, 2)

#35 Vertex 𝑉(−3, 5), axis parallel to the 𝑥-axis, and passing through the point (5, 9).

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#36 Find an equation of the parabola that has vertex 𝑉(3, −2), axis parallel to the 𝑥-axis, and

𝑦-intercept (0,1)

#38 Find an equation for the set of points in an 𝑥𝑦-plane that are equidistant from the point 𝑃(7, 0) and

the line with the equation 𝑥 = 1

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Section 11.2 – Ellipses (p.746)

Definition of an Ellipse

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed

points (the foci) in the plane is a positive constant.

𝑎, 𝑏, 𝑐 > 0 (they are distances, they cannot be negative)

The ratio between c and a is called the eccentricity, the

eccentricity, 𝑒 =𝑐

𝑎

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Find the major axis, minor axis, vertices, and foci of the following ellipses and graph them

#2 𝑥2

25+

𝑦2

16= 1

#10 (𝑥+2)2

25+

(𝑦−3)2

4= 1

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Exercise 1: Find the standard equation of the ellipse with

general equation

16𝑥2 + 9𝑦2 + 96𝑥 − 72𝑦 + 144 = 0, then find the

center, vertices and foci of the ellipse.

Exercise 2: Find an equation for the ellipse that satisfies the given conditions: center 𝐶(1, −2) with

horizontal axis length of 4 and vertical axis length of 8 .

#20 Find an equation for the ellipse that has its center at the origin and has vertices 𝑉(0, ±7) and foci

𝐹(0, ±2).

Exercise 2: Find an equation for the ellipse that has foci 𝐹(7, −2) and 𝐹(1, −2) and a vertex at 𝑉(8, −2).

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#25 Find an equation for the ellipse that has its center at the origin, vertices 𝑉(0, ±6) and passes

through the point (3, 2).

#28 Find an equation for the ellipse that has its center at the origin, passing through (2, 8) and (4, 4).

#30 Find an equation for the ellipse that has its center at the origin with eccentricity 4

7 and vertices

𝑉(±7, 0)

#42 Find an equation for the set of points in an 𝑥𝑦-plane such that the sum of the distances from

𝐹(12, 0) and 𝐹′(−12, 0) is 𝑘 = 26 when

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Section 11.3 – Hyperbolas (p.758)

Definition of a hyperbola A hyperbola is the set of all points in a plane, the difference of

whose distances from two fixed points (the foci) in the plane is a

positive constant.

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Find the vertices, the foci, and the equation of the asymptotes of the hyperbola. Sketch its graph, showing

the asymptotes and the foci.

#2 𝑦2

49−

𝑥2

16= 1

Exrcise 1: 𝑥2

16−

𝑦2

49= 1

#12 (𝑥−3)2

25−

(𝑦−1)2

4= 1

#16 Find the standard equation as well as the vertices, the foci, and the equation of the asymptotes of

the hyperbola. Sketch its graph, showing the asymptotes and the foci.

25𝑥2 − 9𝑦2 + 100𝑥 − 54𝑦 + 10 = 0

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#22 Find an equation for the hyperbola that has its center at the origin and foci 𝐹(±8, 0) and vertices

𝑉(±5, 0).

#31 Foci 0, 10F , asymptotes 1

3y x

Exercise 2: Find an equation for the hyperbola that has vertices 𝑉(−2, 5) and 𝑉(−2, 1) as well as a

focus 𝐹(−2, 6).

Exercise 3: Identify the graphs of the following equations as parabola, circle, ellipse or hyperbola. Identify

the orientation (horizontal or vertical)

a. 5𝑥2 − 7𝑦2 + 30𝑥 + 28𝑦 − 18 = 0

b. 3𝑥2 + 3𝑦2 − 12𝑥 + 6𝑦 − 60 = 0

c. 𝑥2 + 6𝑥 − 8𝑦 + 25 = 0

d. 4𝑥2 − 4𝑦2 − 24𝑥 − 16𝑦 − 52 = 0

e. 5𝑥2 + 7𝑦2 + 40𝑥— 28𝑦 + 3 = 0

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Section 11.5 – Polar Coordinates (p.784)

Instead of expressing points in the coordinate plane by distance in x- and y- direction, as ,P x y , points

are expressed by the radius of a circle with center at the origin O (the pole) and the angle between the

positive x-axis and the line OP. As usual, is positive, if it is measured in counterclockwise direction from

the x-axis. The point is now expressed as ,P r .

The origin or pole has the coordinates 0,O for any .

Polar coordinates are not unique!

Graph the points 𝑃1(4,3𝜋

4) and 𝑃2 (−4,

7𝜋

4)

in the given polar coordinate system:

⟹ polar coordinates can have a negative radius.

This means, we start measuring the angle from

the negative 𝑥 − axis

#1 Which polar coordinates represent the same point as (3,𝜋

3)?

(a) (3,7𝜋

3) (b) (3, −

𝜋

3)

(c) (−3,4𝜋

3) (d) (3, −

2𝜋

3)

(e) (−3, −2𝜋

3) (f) (−3, −

𝜋

3)

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Relationship between rectangular and polar coordinates:

If 𝑃(𝑥, 𝑦) and 𝑃(𝑟, 𝜃) determine the same point 𝑃, then

(1) 𝑥 = 𝑟 cos 𝜃 , 𝑦 = 𝑟 sin 𝜃

(2) 𝑟2 = 𝑥2 + 𝑦2 , tan 𝜃 =𝑦

𝑥 𝑖𝑓 𝑥 ≠ 0

Changing polar coordinates into rectangular coordinates

#4 (a) (5,5𝜋

6) (b) (−6,

7𝜋

3)

#8 (10, arccos (−1

3))

Change the rectangular coordinates to polar

coordinates with 𝑟 > 0 𝑎𝑛𝑑 0 ≤ 𝜃 ≤ 2𝜋

#10 (a) (3√3, 3)

#12 (b) (−4, 4√3)

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Solving polar equations

We use the relationship between rectangular and polar coordinates (𝑥 = 𝑟 𝑐𝑜𝑠𝜃, 𝑦 = 𝑟 sin 𝜃 ) to

transform an equation in x and y into a polar equation.

Find a polar equation that has the same graph as the equation in 𝑥 and 𝑦.

#14 𝑦 = 2

#18 𝑥2 + 𝑦2 = 2

#20 𝑥2 = 8𝑦

#24 2𝑦 = −𝑥 + 4

#30 𝑥2 − 𝑦2 = 9

#34 𝑥2 + (𝑦 − 1)2 = 1

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#36 (𝑥 − 3)2 + (𝑦 + 4)2 = 25

Find an equation in 𝑥 and 𝑦 that has the same graph as the polar equation.

#38 𝑟 sin 𝜃 = −2

#44 : 𝑟 − 6 cos 𝜃 = 0

#48 𝑟2(cos2 𝜃 + 4 sin2 𝜃) = 16

#58 𝑟 = 2 cos 𝜃 − 4 sin 𝜃