Ellipses and Circles

Section 10.3

1st Definition of a Circle

A circle is a conic section formed by a plane intersecting one cone perpendicular to the axis of the double-napped cone.The degenerate conic section that is associated with a circle is a point.

2nd Definition of a Circle

A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius of the circle, and the given point is the center of the circle.

Standard form of a circle with center C (h, k) and radius r is  2 2 2( ) ( )x h y k r

Example 1

Express in standard form the equation of the circle centered at (-2, 3) with radius 5.

2 2 2( ) ( )x h y k r

2 2 2( 2) ( 3) 5x y

2 2( 2) ( 3) 25x y

Example 2

Express in standard form the equation of the circle with center at the origin and radius of 4. Sketch the graph.

2 2 2( ) ( )x h y k r 2 2 2( 0) ( 0) 4x y 2 2 16x y

x

y

Example 3

Find the center and radius of the circle with the equation

Center: Radius =

2 2( 5) ( 1) 32x y

4 2 5, 1

Example 4Write the equation for each circle described below.a. The circle has its center at (8, -9) and

passes through the point at (4, -6).2 2 2( ) ( )x h y k r

2 2 2(4 8) ( 6 9) r 216 9 r

225 r2 2( 8) ( 9) 25x y

b. The endpoints of a diameter are at (1, 8) and (1, -4).

1 1 8 4center : ,2 2

1, 2

2 2 2(1 1) (8 2) r 20 36 r

2 2( 1) ( 2) 36x y

End of 1st Day

1st Definition of an Ellipse

An ellipse is a conic section formed by a plane intersecting one cone not perpendicular to the axis of the double-napped cone.The degenerate conic section that is associated with an ellipse is also a point.

2nd Definition

An ellipse is the set of all points (x, y) in a plane, the sum of whose distances from two distinct fixed points (foci) is constant.d1 + d2 = constant

d1 d2

The line through the foci intersects the ellipse at two points, called vertices. The chord joiningthe vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicularto the major axis at the center is the minor axis of the ellipse.

vertexvertex center

major axis minor axis

General Equation of an Ellipse

Ax2 + Cy2 + Dx + Ey + F = 0If A = C, then the ellipse is a circle.

Standard Equation of an Ellipse

The standard form of the equation of an ellipse, with center (h, k) and major and minor axes of lengths 2a and 2b respectively, where 0 < b < a,

2 2

2 2 1x h y k

a b

where the major axis is horizontal.

2 2

2 2 1x h y k

b a

where the major axis is vertical.

The foci lie on the major axis, c units from the center, with c2 = a2 – b2.

The eccentricity of an ellipse is

cea

Example 1

Find the center, vertices, the endpoints of the minor axis, foci, eccentricity, and graph for the ellipses given in standard form.

a =b = c =

2 2

181 16x y

2 29 4 65

49

center:

vertices:

endpoints of the minor axis:

foci:

eccentricity:

0, 0

9, 0

0, 4

65, 0

659

x

y

V1 F1F2

V2C

Example 2For the following ellipse, find the center, a, b, c, vertices, the endpoints of the minor axis, foci, eccentricity, and graph.

16x2 + y2 − 64x + 2y + 49 = 0What must you do to the general equation above to do this example?Complete the square twice.

2 216 4 4 2 1 49 64 1x x y y

2 216 2 1 16x y

2 22 11

1 16x y

16x2 + y2 − 64x + 2y + 49 = 0

2 216 4 ___ 2 ___ 49 ___ ___x x y y

16x2 − 64x + y2 + 2y = −49

center:vertices:endpoints of the minor axis:

foci:

eccentricity:

2, 1 15 , 2, 1 15

(2, 3), (2, −5) (3, −1), (1, −1)

(2, −1)

4 16 1 15 a = b = c

What type of ellipse is this ellipse? vertical ellipse?

1

154

x

y

V1

V2

F1

F2

C

Example 3

Write the equation of each ellipse in standard form.A. Endpoints of the major axis are at (0, ±10)

and whose foci are at (0, ±8).center: (0, 0)vertical ellipsea = 10; c = 8b = 6

2 2

136 100x y

B. The endpoints of the major axis are at (10, 2) and (–8, 2). The foci are at (6, 2) and (–4, 2).

10 8 2 2center : , 1, 22 2

10 1 9a 6 1 5c

horizontal ellipse

2b 81 25 56

2 21 21

81 56x y

C. The major axis is 20 units in length and parallel to the y-axis. The minor axis is 6 units in length. The center is located at (4, 2).

2 20, 10a a 2 6, 3b b

2 24 21

9 100x y

vertical ellipse