If a hyperbola passes through a focus of the ellipse `(x^2)/(25)+(y^2)/(16)=1` and its transverse and conjugate axes coincide with major and minor axes of the ellipse, and the product of their eccentricityis 1, then

Options

A. `(x^(2))/(9)-(y^(2))/(16)=1`
B. `(x^(2))/(16)-(y^(2))/(9)=-1`
C. `(x^(2))/(9)-(y^(2))/(25)=1`
D. none of these

Explanation

Wehave, `e_(1)=sqrt(1-(16)/(25))=(3)/(5)`
`:.e_(1)e_(2)=1impliese_(2)=(5)/(3)`
The coordinates of foci of the ellipse are `(0,+-3)`.
Clearly, hyperbola in option `(b)` passes through `(0,+-3)` and has eccentricity `(5)/(3)`.