Ellipse and Conic Sections Ellipses

Chapter 13_Graphing the Conic Sections
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 =