Question

The coordinates of the point of intersection of tangents drawn to $y^2=4ax$ at the point of intersection with the line $x\cos \alpha +y\sin \alpha -p=0$ is

• A. $(p\tan \alpha ,2a\sec \alpha )$
• B. $(-p\sec \alpha ,-2a\tan \alpha )$
• C. $(2a\sec \alpha ,p\tan \alpha )$
• D. $(p\sec \alpha ,2a\tan \alpha )$

Answer 1

The correct option is B $(-p\sec \alpha ,-2a\tan \alpha )$


Let the point of intersection be $(h,k)$

Then equation of chord of contact $\left(AB\right)$: $T=0$

$\Rightarrow yk-2a(x+h)=0$
$\Rightarrow 2ax-yk+2ah=0$

Comparing it with $x\cos \alpha +y\sin \alpha -p=0$

$(h,k)\equiv (-p\sec \alpha ,-2a\tan \alpha )$