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Conic Sections

Problems

Hyperbolas

Axis Rotation

Problem : Is the transverse axis of this hyperbola horizontal or vertical: - + = 1 .

Because the y 2 term is negative, the transverse axis is horizontal.

Problem : Find a , b , and c of the following hyperbola: 5x 2 -3y 2 - 20x + 6y + 2 = 0 .

By completing the square, factoring, and putting the equation in standard form, it is evident that a = , b = , and c = .

Problem : The eccentricity of a hyperbola with center (0, 0) and focus 5, 0) is . What is the standard equation for the hyperbola?

e = , so c = 5 , a = 3 , and b = = 4 . The locations of the focus and the center mean that the transverse axis is horizontal, and the y 2 term is negative. So the standard equation for this hyperbola is - = 1 .

Problem : Find the asymptotes of the hyperbola - = 1 .

a = , b = 2 . The asymptotes are y = x and y = - x .

Problem : The transverse axis of a hyperbola is horizontal. One of the asymptotes is y = x + 5 . What is the standard equation of the hyperbola?

a = 1 , b = 2 , and the center of the hyperbola is (0, 5) . The standard equation, then, is - = 1 .

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