, ,

There are two general equations for a hyperbola.

Horizontal hyperbola equation

Vertical hyperbola 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 .

Rewrite as .

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

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 as .

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

Rewrite the equation as .

Raise to the power of .

Raise to the power of .

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

Subtract from both sides of the equation.

Subtract from .

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

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

Simplify the right side of the equation.

Rewrite as .

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

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.

Subtract from .

The expression contains a division by The expression is undefined.

Undefined

Undefined

The general equation for a vertical hyperbola is .

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

Simplify the numerator.

Multiply by .

Add and .

Raise to the power of .

Simplify the numerator.

Multiply by .

Add and .

Raise to the power of .

Find Hyperbola: Center (0,0), Focus (0,10), Vertex (0,-6) (0,0) , (0,10) , (0,-6)