Question

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

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

Answer 1

Answer: (C)

The focus of the parabola $y^{2}=8$ ax is $(2 a, 0)$.
So, the coordinates of the point on the axis of the parabola at a distance $8 a$ from the focus is $(10 a , 0)$.
Equation of a normal to the parabola $y ^{2}=8 ax$ is
$y=m x-4 a m-2 a m^{3}$
Since it passes through $(10 a , 0), $
$\therefore 0=10 am -4 am -2 am ^{3}$
$\Rightarrow 2 am \left(3- m ^{2}\right)=0 $
$\Rightarrow m ^{2}=3(\because m \neq 0)$
$\Rightarrow m=\pm \sqrt{3}=\tan \left(\pm \frac{\pi}{3}\right)$