Question

If a hyperbola passes through the focii of the ellipse$\frac{x^2}{25}+\frac{y^2}{16}=1.$ Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities hyperbola and ellipse 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 $\left(5\sqrt{3,}0\right)$

Answer 1

The correct option is A the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{16}=1$

Formula,

$e^2=1-\frac{b^2}{a^2}$

$=1-\frac{16}{25}$

$\therefore e=\frac{3}{5}$

$e_2\times e=1$

$\Rightarrow e_2=\frac{5}{3}$

Equation,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Given,

$\Rightarrow (3,0)$

$\frac{3^2}{a^2}=1$

$\Rightarrow a^2=9$

we have,

$e_2^2=1+\frac{b^2}{a^2}$

$\frac{25}{9}=1+\frac{b^2}{9}$

$\Rightarrow b^2=16$

$\frac{x^2}{9}-\frac{y^2}{16}=1$

Hence the required equation.