Question

The portion of a tangent to a parabola $y^2=4ax$ cuts off between the directrix and the curve subtends an angle $\theta$ at the focus, where $\theta =$

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. None of these

Answer 1

Answer: (C)

$\frac{\pi }{2}$

The equation of the tangent at $P(a{t}^2,2at)$ to $y^2=4ax$ is $ty=x+a{t}^2$ ....... $\left(1\right)$

It meets the directrix $x=-a$

$\therefore ty=-a+a{t}^2\Rightarrow y=\frac{a(t^2-1)}{t}$

Thus, $\left(1\right)$ meets the directrix at $Q(-a,\frac{a(t^2-1)}{t})$

Now, slope of $PS$ is $m_1=\frac{2at-0}{a{t}^2-a}=\frac{2t}{t^2-1}$

And slope of $QS$ is $m_2=\frac{a(t^2-1)/(t-0)}{-a-a}=-\frac{(t^2-1)}{2t}$

Since $m_1{m}_2=-1$, therefore $PQ$ subtends a right angle at the focus.

Hence, $\theta =\frac{\pi }{2}$.