If the product of the lengths of the perpendiculars from any point on the hyperbola $16 x^2-25 y^2=400$ to its asymptotes is $p$ and the angle between the two asymptotes is $ heta$, then $p an \frac{ heta}{2}=$

Question

If the product of the lengths of the perpendiculars from any point on the hyperbola $16 x^2-25 y^2=400$ to its asymptotes is $p$ and the angle between the two asymptotes is $	heta$, then $p 	an \frac{	heta}{2}=$

Aicharya · Accepted Answer

Equation of given hyperbola is $ 16 x^2-25 y^2=400 $

Let a point on hyperbola (i), $A(5 \sec \alpha, 4 \tan \alpha)$, so, lengths of the perpendiculars from point $A$ to the asymptotes is $ \frac{20 \sec \alpha+20 \tan \alpha}{\sqrt{16+25}} \text { and } \frac{20 \sec \alpha-20 \tan \alpha}{\sqrt{16+25}} $ So, $\quad P=\frac{400}{41}$ and $\tan \frac{\theta}{2}=\frac{4}{5}$ So, $ P \tan \frac{\theta}{2}=\frac{320}{41} $

If the product of the lengths of the perpendiculars from any point on the hyperbola $16 x^2-25 y^2=400$ to…

Solution