Question

Area of the triangle formed by the lines $x-y=0,x+y=0$ and ant tangent to the hyparabola $x^2-y^2=a^2$ is

• A. $|a|$
• B. $\frac{1}{2}|a|$
• C. $a^2$
• D. $\frac{1}{2}{a}^2$

Answer 1

The correct option is C $a^2$

Equation of line

$x-y=0......\left(1\right)$

$x+y=0.......\left(2\right)$

Equation of hyperbola is

$x^2-y^2=a^2......\left(3\right)$

Let the point $P(a\sec \theta ,a\tan \theta )$ on the hyperbola.

Equation of tangent is,

$x{x}_1-y{y}_1=a^2$

$\Rightarrow a(x\sec \theta -y\tan \theta )=a^2$

$\Rightarrow x\sec \theta -y\tan \theta =a......\left(4\right)$

Now,

Area of $\Delta AOB$ $=\frac{1}{2}$

$=\frac{1}{2}\left|a^2\right({\tan }^2\theta -{\sec }^2\theta )-a^2({\sec }^2\theta -{\tan }^2\theta \left)\right|$

$=\frac{1}{2}|a^2(-1)-a^2\left(1\right)|$

$=\frac{1}{2}|-2a^2|$

$=|-a^2|$

$=\left|a^2\right|$

Hence, this is the answer.