Question

The product of the perpendiculars from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is

• A.
• B.
  
  
• C.
  
  
• D.
  
  

Answer 1

The correct option is C



Let P$(x_1,y_1)$ be any point on $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\Rightarrow \frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}=1$
Product of perpendicular from $(x_1,y_1)$ to asymptotes $\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$ is $\left|\frac{\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}}{\sqrt{(}\frac{1}{a^2}+\frac{1}{b^2}{)}^2}\right|=\frac{1}{\frac{1}{a^2}+\frac{a}{b^2}}=\frac{a^2{b}^2}{a^2+b^2}$