Conic Sections: Hyperbolas
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Definitions
Hyperbolas with Vertices on the x-Axis
A hyperbola is much like an ellipse turned inside out!
The standard form of the equation of a hyperbola formed around the x-axis, with the center at (0, 0) is
.
This is a hyperbola that will never cross the y-axis.
Notice the similarity between this equation and the equation for a horizontal ellipse, which is 
.
Notice that several items are labeled near the hyperbola.
Hyperbolas with Vertices on the y-Axis
Just as ellipses can be oriented horizontally or vertically, hyperbolas can also have a vertical orientation. The standard form of the equation of a hyperbola formed around the y-axis, with the center at (0, 0) is
.
This hyperbola will never cross the x-axis.
Notice the similarity between this equation and the equation for a vertical ellipse 
.
Practice Problems
Problem 1: Your First Bowl on a Potter's Wheel
You have been learning to use a potter's wheel in your art class. You have managed to make a variety of vases and mugs, but now you want to make a bowl. You want the new bowl to be fairly "flat," which is very difficult. The vertex will be 1 inch from the platform on which the clay sits (this is the height of the wheel on the potter's wheel). Adjust b (the y-radius) until you have found a desirable "bowl" shape.
Problem 1 Answers:
As you adjust the size of b, what happens to the shape of the bowl?
Describe the foci and asymptotes of your final "bowl" hyperbola.
Problem 2: Designing a Nuclear Power Plant Cooling Tower
Nuclear power plants often have large cooling towers, with sides curved in a hyperbolic shape. You are designing a tower that will be 40 feet across at the narrowest section, and 100 feet high. Enter the parameters of the cooling tower in the input lines below and graph a hyperbola that will resemble the tower. Keep in mind that the tower will not have a center at (0,0)! Remember that 
represents the center of the hyperbola.
Problem 2 Answers:
Describe approximately where you would find the foci of this hyperbola.
If you change b (the y-radius of the hyperbola) from 20 feet to 30 feet, what happens to the shape of the cooling tower?
With 
describe what happens to the foci of the hyperbola.
Do you believe that increasing b from 20 to 30 feet will improve the performance of the cooling tower? Explain your answer.
