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 :

• A. the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{16}=1$
• B. the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{25}=1$
• C. focus of hyperbola is $($5, $0)$
• D. focus of hyperbola is $(5\sqrt{3},0)$

Answer 1

Answer: (A), (C)

focus of hyperbola is $($5, $0)$
the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{16}=1$

Eccentricity of ellipse $=\frac{3}{5}$
Eccentricity of hyperbola $=\frac{5}{3}$ and it passes through $(\pm 3,0)$
$\Rightarrow$ its equation is $\frac{x^2}{9}-\frac{y^2}{b^2}=1$
where $1+\frac{b^2}{9}=\frac{25}{9}\Rightarrow b^2=16$
$\Rightarrow \frac{x^2}{9}-\frac{y^2}{16}=1$ and its foci are $(\pm 5,0)$.