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Reviewed CSCA Mathematics question · Hard

A hyperbola centered at the origin has a vertex at (0, -4) and a focus at (0, -6). What is the length of its conjugate axis?

  1. 4√5
  2. 2√5
  3. 8√5
  4. 10
Show the answer and explanation

Correct answer

A. 4√5

Principle or equation

For a vertical hyperbola centered at origin, the distance from center to vertex is a, to focus is c, and c^2 = a^2 + b^2. The conjugate axis length is 2b.

Why this answer is correct

Vertex at (0, -4) gives a = 4. Focus at (0, -6) gives c = 6. Then b^2 = c^2 - a^2 = 36 - 16 = 20, so b = 2√5. Conjugate axis length = 2b = 4√5.

Example

For a hyperbola with a = 3 and c = 5, b = 4, conjugate axis = 8.

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