Hyperbola
Eccentricity Relations
JEE Main 2023
Grade 11
Question:
If the eccentricities of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the ellipse $\frac{x^2}{p^2} + \frac{y^2}{q^2} = 1$ satisfy $e_H \cdot e_E = 1$, and both share the same foci, then which is correct?
(1) $a^2 + b^2 = p^2 + q^2$
(2) $a^2 - b^2 = p^2 - q^2$
(3) $a^2 + p^2 = b^2 + q^2$
(4) $a^2b^2 = p^2q^2$
Step-by-Step Solution
Key Concept: Same foci means $c_H = c_E = c$. For hyperbola $c^2 = a^2 + b^2$; for ellipse $c^2 = p^2 - q^2$. So $a^2+b^2 = p^2-q^2...$ that gives $a^2+b^2+q^2 = p^2$. Also $e_H \cdot e_E = 1 \Rightarrow (c/a)(c/p) = 1 \Rightarrow c^2 = ap$. Use these to check options.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)