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Q.
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
NTA AbhyasNTA Abhyas 2022
Solution:
Equation of the tangent to the hyperbola $x^{2}-y^{2}=16$ in parametric form is
$4xsec \phi-4ytan \phi = 16$
$\Rightarrow xsec \phi - y tan \phi = 4$
$A(4(\sec \phi+\tan \phi), 4(\sec \phi+\tan \phi))$
$B(4(\sec \phi-\tan \phi), 4(\tan \phi-\sec \phi))$
$\therefore$ Area bounded by $\Delta A O B$
$=\left|\frac{1}{2}\left(16\left(\tan ^{2} \phi-\sec ^{2} \phi\right)-16\left(\sec ^{2} \phi-\tan ^{2} \phi\right)\right)\right|$
$=\left|\frac{1}{2}(-16-16)\right|=16$ sq. units