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, the product of their eccentricities being $1$ than:

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

Answer 1

Answer: (A), (C)

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

Let the equation of the hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b}=1$ ...(1)
Since it passes through the focus $(\pm \sqrt{25-16},0)=(\pm 3,0)$
of the ellipse, so $\frac{9}{a^2}=1\Rightarrow a^2=9$ ...(2)
Also product of this eccentricities =1
$\Rightarrow \sqrt{1-\frac{16}{25}}\times \sqrt{1+\frac{b^2}{a^2}}=1$
$\Rightarrow 1+\frac{b^2}{9}=\frac{25}{9}\Rightarrow b^2=16$
Therefore equation of hyperbola will be $\frac{x^2}{9}-\frac{y^2}{16}=1$
Its focus $=(\pm \sqrt{a^2+b^2,}0)=(\pm 5,0)$