Question

If the normal drawn from the point on the axis of the parabola $y^2=8ax$ whose distance from the focus is $8a$, and which is not parallel to either axes, makes an angle $\theta$ with the axis of $x$, then $\theta$ is equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is D $\frac{\pi }{3}$

Focus is $(2a,0)$
So point is $P(2a+8a,0)$
Equation of any normal is $y=mx-2\left(2a\right)m-2a{m}^3$
If it passes through $P(10a,0)$, then
$6a-2a{m}^3=0\Rightarrow m=0,\pm \sqrt{3}$
So the required angle is $\theta ={\tan }^{-1}\sqrt{3}=\frac{\pi }{3}$