Question

From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$. The area cut off by the chord of contact on the region between the asymptotes is equal to.

• A. $\frac{ab}{2}$
• B. ab
• C. $2ab$
• D. $4ab$

Answer 1

The correct option is D $4ab$

$x_1=\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$x_2:\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$

Eqn of chord of contact from $(a,,0)$ to $H_2$:

$\frac{ax}{a^2}-\frac{ay}{b^2}=2$

Area cut off by chord of contact

$A_1:ay=bx$ $y=-\frac{b}{a}x$

$A_2:ay=-bx$ $y=\frac{b}{a}x$

$P=(2a,2,b)$

$Q=(2a,-2b)$