Question

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :

• A. \( \frac{13}{6} \)
• B. 2
• C. \( \frac{13}{12} \)
• D. \( \frac{13}{8} \)

Hint:

Use \( 2b = 5 \), \( 2ae = 13 \) and \( b^2 = a^2(e^2-1) \) to solve for \( e \).

Answer 1

The Correct Option is C

Conjugate axis \( 2b = 5 \Rightarrow b = \frac{5}{2} \). Distance between foci \( 2ae = 13 \Rightarrow a = \frac{13}{2e} \).
For a hyperbola \( b^2 = a^2(e^2-1) \): \( \frac{25}{4} = \frac{169}{4e^2}(e^2-1) \).
\( 25e^2 = 169e^2 - 169 \Rightarrow 144e^2 = 169 \Rightarrow e = \frac{13}{12} \).
Answer: (3) \( \frac{13}{12} \)