Question

If tangent at $P$ and at vertex of the parabola $y^2=4ax$ meets at $Q$, then the value of $\angle SQP$, where $S$ is the focus, is

• A. $\frac{\pi }{2}$
• B. $\pi$
• C. $\frac{3\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is A $\frac{\pi }{2}$

Given : $y^2=4ax$
Let any point on parabola be $P(a{t}^2,2at)$
Tangent at $P$
$ty=x+a{t}^2\cdots \left(1\right)$
And tangent at vertex is
$x=0\cdots \left(2\right)$
Solving $\left(1\right)$ and $\left(2\right)$, we get
$\Rightarrow y=at$
So, the coordinates are
$Q\equiv (0,at)\text{and}S=(a,0)$
Now, slopes are
$m_{QP}=\frac{2at-at}{a{t}^2-0}=\frac{1}{t}{m}_{QS}=\frac{0-at}{a-0}=-t\Rightarrow m_{QP}\times m_{QS}=-1\therefore \angle SQP=\frac{\pi }{2}$