Analyze the Equation of an Ellipse

Example: Find the center, foci, and vertices of the ellipse. Graph the equation.

(x-1)^2/25+(y+1)^2/64=1

Solution:

Finding the center:

There are two standard orientations of ellipse’s, both of them have the center at (h,k).

Horizontal: (x-h)^2/a^2+(y-k)^2/b^2=1

Vertical: (x-h)^2/b^2+(y-k)^2/a^2=1

For this example the center of the ellipse is (1,-1)

Finding the orientation:

The larger denominator will indicate the orientation. Since our example has a larger denominator under the y variable the orientation is vertical. That means the ellipse is longer vertically.

Finding the vertices: (Major Axis)

a is the distance between the center and the vertex along the major or longer axis. Since our larger denominator is 64 a is 8. I know this because the in the formula the larger denominator is a^2 . If a^2=64 then a=8 .

Since the ellipse is vertically oriented the vertices will be 8 units above and below the center.

This puts the vertices at (1,7) and (1,-9).

Finding the vertices: (Minor Axis)

b is the distance between the center and the vertex along the minor or shorter axis. Since our smaller denominator is 25 b is 5. I know this because the in the formula the smaller denominator is b^2 . If b^2=25 then b=5 .

Since the ellipse is vertically oriented the vertices along the minor axis or shorter axis will be 5 units left and right of the center.

This puts the vertices at (-4,-1) and (6,-1).

Finding the foci:

c is the distance between the center and the focus. You can not find c from the equation. You must know that a, b, and c are related in a similar formula to Pythagorean theorem.

b^2= a^2-c^2