, ,

There are two general equations for an ellipse.

Horizontal ellipse equation

Vertical ellipse equation

Use the distance formula to determine the distance between the two points.

Substitute the actual values of the points into the distance formula.

Simplify.

Subtract from .

Raising to any positive power yields .

Subtract from .

Raise to the power of .

Add and .

Use the distance formula to determine the distance between the two points.

Substitute the actual values of the points into the distance formula.

Simplify.

Subtract from .

Raising to any positive power yields .

Subtract from .

Raise to the power of .

Add and .

Rewrite the equation as .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

Multiply each term in by

Multiply each term in by .

Multiply .

Multiply by .

Multiply by .

Multiply by .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the to find the first solution.

Next, use the negative value of the to find the second solution.

The complete solution is the result of both the positive and negative portions of the solution.

is a distance, which means it should be a positive number.

Slope is equal to the change in over the change in , or rise over run.

The change in is equal to the difference in x-coordinates (also called run), and the change in is equal to the difference in y-coordinates (also called rise).

Substitute in the values of and into the equation to find the slope.

Simplify.

Multiply by .

Subtract from .

The expression contains a division by The expression is undefined.

Undefined

Undefined

The general equation for a vertical ellipse is .

Substitute the values , , , and into to get the ellipse equation .

Multiply by .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Multiply by .

Factor out of .

Separate fractions.

Divide by .

Divide by .

Multiply by .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Multiply by .

Factor out of .

Separate fractions.

Divide by .

Divide by .

Find Ellipse: Center (5,0.12), Focus (5,7), Vertex (5,22) (5,0.12) , (5,22) , (5,7)