$\text{ If }PQ\text{ is a double ordinate of the hyperbola }16x^2-25y^2=400\text{ be produced on both }$$\text{ sides to meet the asymptotes at }R\text{ and }S,\text{ then }PR\cdot PS=$

$\begin{array}{r}\text{ Given hyperbola is }\\ 16x^2-25y^2=400\end{array}$

$\Rightarrow \frac{x^2}{25}-\frac{y^2}{16}=1$

$\text{ Let }P(a\sec \theta ,b\tan \theta )\text{ be a point on hyperbola }$

$\Rightarrow Q(a\sec \theta ,-b\tan \theta )\text{ (Image of }P\text{ with respect to }x\text{ -axis) }$

$\text{ Equation of }PQ\text{ is }x=a\sec \theta$

$\text{ It intersects with asymptotes }\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$

$\Rightarrow y=\pm b\sec \theta$

$\begin{array}{l}\text{ So, }R(a\sec \theta ,b\sec \theta ),S(a\sec \theta ,-b\sec \theta )\\ \therefore PR\cdot PS=b^2\left({\sec }^2\theta -{\tan }^2\theta \right)\\ =b^2\\ =16\end{array}$