Hyperbola
Eccentricity Relations
JEE Main
Grade 11
Question:
If a hyperbola passes through the focus of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is $1$, then the length of the latus rectum of the hyperbola is _____.
Step-by-Step Solution
Key Concept: Ellipse: $a = 5, b = 4, c = 3, e_E = 3/5$. Hyperbola: $e_H = 5/3$, so $b_H^2/a_H^2 = e_H^2 - 1 = 16/9$. Passes through $(3, 0)$: $9/a_H^2 = 1 \Rightarrow a_H = 3, b_H^2 = 16$. Latus rectum $= 2b_H^2/a_H = 32/3$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: \frac{32}{3}