16. 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 $( 5 \sqrt { 3 } , 0 )$

\section*{Sol. (A), (C)}
Eccentricity of ellipse $= \frac { 3 } { 5 }$\\
Eccentricity of hyperbola $= \frac { 5 } { 3 }$ and it passes through $( \pm 3,0 )$\\
⇒ its equation $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$\\
where $1 + \frac { \mathrm { b } ^ { 2 } } { 9 } = \frac { 25 } { 9 } \Rightarrow \mathrm {~b} ^ { 2 } = 16$\\
$\Rightarrow \quad \frac { \mathrm { x } ^ { 2 } } { 9 } - \frac { \mathrm { y } ^ { 2 } } { 16 } = 1$ and its foci are $( \pm 5,0 )$.\\