A hyperbola centered at the origin has a vertex at (-3, 0) and a focus at (-5, 0). What is the length of its conjugate axis?
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Correct answer
C. 8
Principle or equation
For a hyperbola centered at the origin with transverse axis along the x-axis, the standard equation is x^2/a^2 - y^2/b^2 = 1. The distance from the center to a vertex is a, to a focus is c, and they satisfy c^2 = a^2 + b^2. The conjugate axis has length 2b.
Why this answer is correct
The vertex is at (-3,0), so a = 3. The focus is at (-5,0), so c = 5. Using c^2 = a^2 + b^2, we have 25 = 9 + b^2, so b^2 = 16, hence b = 4. The conjugate axis length is 2b = 8.
Example
For a hyperbola with a = 4 and c = 5, b = sqrt(25 - 16) = 3, so the conjugate axis length is 6.
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