Question

The portion of the tangent intercepted between the point of contact and the directrix of the parabola \( y^2 = 4ax\) subtends at the focus an angle of

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is D ${90}^{\circ }$

If $P=(a{t}^2,2at)$ then the equation of tangent at t is $ty=x+a{t}^2.......\left(1\right)$
Directrix is $x+a=0............\left(2\right)$
By solving $Q=[-a,\frac{a(t^2-1)}{t}],S=(a,0)$
Slope of $SP=\frac{2at}{a(t^2-1)}=\frac{2t}{t^2-1}$,Slope of $SQ=\frac{\frac{a(t^2-1)}{t}}{-2a}=\frac{-(t^2-1)}{2t}$
Product of slopes = –1. $\therefore \angle PSQ={90}^{\circ }$