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 $\theta$, then $p \tan \frac{\theta}{2}=$

• A

  $\frac{400}{41}$
• Correct answer
  B

  $\frac{320}{41}$
• C

  $\frac{4}{5}$
• D

  $\frac{25}{16}$

Solution

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, $$ P \tan \frac{\theta}{2}=\frac{320}{41} $$

Hence, the correct answer is option: 'B'.