Q. The tangent drawn at any point $P$ to the parabola $y^{2} = 4ax$ meets the directrix at the point $K$, then the angle which $KP$ subtends at its focus is
A) $30^{{^\circ}}$
B) $45^{{^\circ}}$
C) $60^{{^\circ}}$
D) $90^{{^\circ}}$
Explanation
(d)
Let $P\left( at^{2},\ 2at \right)$ any point on the parabola and focus is $(a,\ 0)$
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The equation of tangent at $P$ is $yt = x + at^{2}$
Since, it meets the directrix $x = - a$ at $K$
Then, the coordinate of $K$ is $\left( - a,\ \frac{at^{2} - a}{t} \right)$
Slope of $SP = m_{1} = \frac{2at}{a\left( t^{2} - 1 \right)}$
Slope of $SK = m_{2} = \frac{a\left( t^{2} - 1 \right)}{- 2at}$
$\therefore m_{1}m_{2} = \frac{2at}{a\left( t^{2} - 1 \right)} \bullet \frac{a\left( t^{2} - 1 \right)}{( - 2at)} = - 1$
$\therefore\angle PSK = 90^{{^\circ}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.