6-4 Hyperbolas

The standard form of a hyperbola:

The Steps for solving a hyperbola:

• The larger denominator is the major axis.
• The smaller denominator is the minor axis.
• Vertex = the square root of the largest denomiator
• Other value = the square root of the smallest denomiator
• Find the focus with the equation f to the second = larger denominator + smaller denominatior. Then f to the second = vertex to the second + the other value to the second.
• Find the asymptotes. Equation: y= +- the square root of the y denominator over the square root of the x denominator.

Example:
Solve the hyperbola.

9x² - 16y² = 144

First, you divide everything by 144. Which will leave you with:
x²/16 - y²/9 = 1

After you divided everything by 144, you officially start the steps from above. You need to state what is your major and minor axis.The major axis has to be positive. One will be negative (either x or y, it doesn't matter).
Major axis: x
Minor axis: y

After you have found the major and minor axis, you need to take the square root of your major axis.
Which would be the square root of 16 = +-4. Your points will be (4,0) (-4,0).

Then you do the same thing for your minor axis. Square root of 9 = +-3. Points: (0,3) (0,-3).

Now you have to find your focus. To find your focus, you use the equation focus squared = larger denominator + smaller denominator. F^2 = 16 + 9 = 25. You take the square root of your answer which is 25. So the square root of 25 = +-5. Focus: (5,0) (-5,0).

Your last step is to find the asymptotes. You use the equation in the last step from the steps above that I listed. Y = square root of 9/square root of 16 = 3/4x. Your finally answer for your asymptote is Y = +-3/4x.

Amber Smith