# Precalculus Examples

Step 1

Simplify each term in the equation in order to set the right side equal to . The standard form of an ellipse or hyperbola requires the right side of the equation be .

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

Step 3

Match the values in this ellipse to those of the standard form. The variable represents the radius of the major axis of the ellipse, represents the radius of the minor axis of the ellipse, represents the x-offset from the origin, and represents the y-offset from the origin.

Step 4

The center of an ellipse follows the form of . Substitute in the values of and .

Step 5

Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

Step 5.2

Substitute the values of and in the formula.

Step 5.3

Simplify.

Step 5.3.1

One to any power is one.

Step 5.3.2

Apply the product rule to .

Step 5.3.3

One to any power is one.

Step 5.3.4

Raise to the power of .

Step 5.3.5

Write as a fraction with a common denominator.

Step 5.3.6

Combine the numerators over the common denominator.

Step 5.3.7

Subtract from .

Step 5.3.8

Rewrite as .

Step 5.3.9

Simplify the denominator.

Step 5.3.9.1

Rewrite as .

Step 5.3.9.2

Pull terms out from under the radical, assuming positive real numbers.

Step 6

Step 6.1

The first vertex of an ellipse can be found by adding to .

Step 6.2

Substitute the known values of , , and into the formula.

Step 6.3

Simplify.

Step 6.4

The second vertex of an ellipse can be found by subtracting from .

Step 6.5

Substitute the known values of , , and into the formula.

Step 6.6

Simplify.

Step 6.7

Ellipses have two vertices.

:

:

:

:

Step 7

Step 7.1

The first focus of an ellipse can be found by adding to .

Step 7.2

Substitute the known values of , , and into the formula.

Step 7.3

Simplify.

Step 7.4

The second focus of an ellipse can be found by subtracting from .

Step 7.5

Substitute the known values of , , and into the formula.

Step 7.6

Simplify.

Step 7.7

Ellipses have two foci.

:

:

:

:

Step 8

Step 8.1

Find the eccentricity by using the following formula.

Step 8.2

Substitute the values of and into the formula.

Step 8.3

Simplify.

Step 8.3.1

Divide by .

Step 8.3.2

One to any power is one.

Step 8.3.3

Apply the product rule to .

Step 8.3.4

One to any power is one.

Step 8.3.5

Raise to the power of .

Step 8.3.6

Write as a fraction with a common denominator.

Step 8.3.7

Combine the numerators over the common denominator.

Step 8.3.8

Subtract from .

Step 8.3.9

Rewrite as .

Step 8.3.10

Simplify the denominator.

Step 8.3.10.1

Rewrite as .

Step 8.3.10.2

Pull terms out from under the radical, assuming positive real numbers.

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center:

:

:

:

:

Eccentricity:

Step 10