Ellipses

In this study guide we will focus on graphing ellipses but be sure to read and understand the definition in your text.

Equation of an Ellipse (standard form)

Area of an Ellipse

( x − h) 2 ( y − k ) 2

+

=1

a2

b2

with a horizontal axis that measures 2a units, vertical axis measures 2b units, and (h, k) is the center.

The long axis of an ellipse is called the major axis and the short axis is called the minor axis. These axes terminate at points that we will call vertices. The vertices along the horizontal axis will be ( h ± a, k ) and the vertices along the vertical axes will be ( h, k ± b) . These points, along with the center, will provide us with a method to sketch an ellipse given standard form.

A = π ab

Graph

( x − 5) 2 ( y − 8) 2

+

=1

9

25

First plot the center.

Then use a = 3 and

plot a point 3 units to

the left and 3 units to

the right of the

center.

Use standard form to

identify a, b, and the

center (h, k).

Next, use b = 5 and

plot a point 5 units up

and 5 units down

from the center.

Label at least 4

points on the ellipse.

In this example the major axis is the vertical axis and the minor axis is the horizontal axis. The major axis measures 2b = 10 units in length and the minor axis measures 2a = 6 units in length. There are no x- and y- intercepts in this example.

Problems Solved!

13.4 - 1

Chapter 13_Graphing the Conic Sections

Ellipses

A. Graph the ellipse. Label the center and 4 other points.

x2 y2

+

=1

36 4

( x − 2) 2 y 2

+

=1

64

25

( x − 2) 2 ( y + 3) 2

+

=1

4

16

Problems Solved!

13.4 - 2

Chapter 13_Graphing the Conic Sections

Ellipses

It will often be the case that the ellipse will not be given in standard form. In this case we will have to rewrite the equation in standard form first.

B. Graph the ellipse. Label the center and 4 other points.

64 x 2 + 4 y 2 = 256

20( x + 10) 2 + 9 y 2 = 180

5( x + 1) 2 + 7( y − 2) 2 = 35

When graphing ellipses be sure that they do not look like diamonds. Round them up as best you can by hand but keep in mind that they do not have to be perfect. Problems Solved!

13.4 - 3

Chapter 13_Graphing the Conic Sections

Ellipses

The next examples require us to complete the square to obtain standard form. Remember 2

to factor the leading coefficient out of each variable grouping before using ( B ) to 2

complete the square.

C. Graph the ellipse. Label the center and 4 other points.

4 x 2 + 9 y 2 + 32 x − 90 y + 253 = 0

Factor the leading

coefficient out of

each variable

grouping before

using

( B )2

2

5 x 2 + 3 y 2 − 10 x − 12 y + 2 = 0

Problems Solved!

13.4 - 4

Chapter 13_Graphing the Conic Sections

Ellipses

D. Answer the questions.

Find the x- and y-intercepts for the ellipse

( x − 1) 2 ( y − 2) 2

+

=1

3

5

Find the x- and y-intercepts for the ellipse

( x + 10) 2 y 2

+

=1

9

20

When the calculation for intercepts yields complex results then there are no real intercepts. This just means that the graph does not cross that particular axis.

Problems Solved!

13.4 - 5

Chapter 13_Graphing the Conic Sections

Ellipses

Find the equation of the ellipse centered at (3, -1) with a horizontal major axis of length 10 units and vertical minor axis of length 4 units.

Find the equation of the ellipse given the vertices (±3, 0) and (0, ± 8)

Find the equation of the ellipse whose major axis has vertices (-1, -2) and (-1, 10) and minor axis has co-vertices (-3, 4) and (1, 4).

Problems Solved!

13.4 - 6