Ellipses and Circles

Section 10.3

1st Definition

An ellipse is a conic section formed by a plane intersecting one cone not perpendicular to the axis of the double-napped cone.A circle is a conic section formed by a plane intersecting one cone perpendicular to the axis of the double-napped cone.

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

Turn on N-Spire Calculator.Open the file Ellipse Construction.

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 1

x h y k

a bwhere the major axis is horizontal.

2 2

2 2 1

x h y k

b awhere the major axis is vertical.

The foci lie on the major axis, c units from the center, with c2 = a2 – b2.What is true about c in a circle? Why?It is equal to 0.Because a and b are equal lengths.What is true about the center and foci of a circle?They are all the same point.To measure the ovalness of an ellipse, you can use the concept of eccentricity.

Eccentricity ofan Ellipse

The eccentricity of an ellipse is given by the ratio

Note that 0 < e < a for every ellipse. Why?c < a

The closer that the eccentricity is to 1 the more elongated the ellipse.What is the eccentricity of a circle?

0

cea

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

What must you do to the above equation to do this example?Complete the square twice.

16x2 + y2 − 64x + 2y + 49 = 0

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

center:1 4 16 1 15

(2, -1) a = b = c

2 216 2 1 16x y

2 22 11

1 16x y

What type of ellipse is this ellipse?

vertical ellipse?

vertices:endpoints of the minor axis:

eccentricity:

foci: 2, 1 15 , 2, 1 15

154

e

(2, 3), (2, −5) (3, −1), (1, −1)

x

y

V1

V2

F1

F2

C

154

e

A circle is a special ellipse. The center and the two foci are the same point.A circle is a set of points in a plane a given distant from a given point.The standard form of the equation of a circle with center (h, k) and radius, r is

(x – h)2 + (y – k)2 = r2

Example

Find the standard form of the equation of the circle, center, radius and graph.

2 28 18 72 x x y y

(x + 4)2 + (y – 9)2 = 25center: (-4, 9)radius = 5

2 28 ___ 18 ___ 72 ___ ___ x x y y

22 2 28 4 18 9 72 16 81 x x y y

END OF THE 1ST DAY

Each focus has its own line that relates to the ellipse, this line is called the directrix. If we have an ellipse with a major axis distance of a and an eccentricity of e, the directrix of the ellipse is defined as the lines perpendicular to the line containing the major axis at a distance from the   center of .a

e

Because the eccentricity of an ellipse is positive and less than 1, we know that

and therefore we know that the directrix does not intersect the ellipse.

a ae

Why?

Directrix Directrixa

ae

ae

Focus Directrix Property of Ellipses

This property explains how the directrix relates to an ellipse. This is the 3RD DEFINITION OF AN ELLIPSE.

An ellipse is the set of all points P such that the distance from a point on the ellipse to the focus F is e times the distance from the same point to the associated directrix.

d2d1

P

F1

Directrix Directrix

A

F1P = e • AP

Example

Given: a vertical ellipse with

Find the length from P(2, 4), a point on the ellipse to the focus associated with the given directrix.

3 and the5

e 37equation of a directrix: .3

y

37 43

AP

253

AP

F1P = e • AP

13 255 3

F P

1 5F P

Examples

Write the equation of each ellipse described. Find the equation of each directrix. Graph.

1. Center (0, 0), a = 6, b = 4 horizontal major axis.

To find the equation of the directrix. a. Find c.

b. Find e.

2 2

136 16

x y

36 16 2 5 c

2 5 56 3

cea

c. Find .ae6 18 5

553

ae

d. Add and subtract from the appropriate coordinate

of the center.

ae

18 55

x

x

y

3. Center (6, 1), foci (6, 5) and (6, −3)length of major axis is 10

vertical major axis2c = 5 + 3 = 8 so c = 416 = 25 – b2

b2 = 9

2 26 11

9 25x y

1 17 and 54 4

y y

x

y

x

y

x

y

x

y

x

y

x

y

x

y