Question

The area of triangle formed by the lines $x-y=0,x+y=0$ and any tangent to the hyperbola $x^2-y^2=16$ is equal to

• A. $2$ sq. units
• B. $4$ sq. units
• C. $8$ sq. units
• D. $16$ sq. units

Answer 1

Answer: (D)

Equation of the tangent to the hyperbola $x^2-y^2=16$ in parametric
form is
$4x\sec \varphi -4y\tan \varphi =16$
$\Rightarrow x\sec \varphi -y\tan \varphi =4$
Solving with $y=x$ and $y=-x,$ we get,
$A(4(\sec \varphi +\tan \varphi ),4(\sec \varphi +\tan \varphi ))$
$B(4(\sec \varphi -\tan \varphi ),4(\tan \varphi -\sec \varphi ))$
$\therefore$ Area bounded by $\Delta AOB$ $=\mid \frac{1}{2}\left(16\left({\tan }^2\varphi -{\sec }^2\varphi \right)-16\left({\sec }^2\varphi -{\tan }^2\varphi \right)\right)$
$=\left|\frac{1}{2}(-16-16)\right|=16sq.$ units