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 axis, makes an angle $\theta$ with 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

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

The focus of the parabola $y^2=8ax$ is $(2a,0)$.

So, the coordinates of the point on the axis of the parabola at a distance $8a$ from the focus is $(10a,0)$.

Equation of a normal to the parabola $y^2=8ax$ is

$y=mx-4am-2a{m}^3$

Since it passes through $(10a,0)$

$\therefore 0=10am-4am-2a{m}^3\Rightarrow 2am(3-m^2)=0\Rightarrow m^2=3(\because m\ne 0)$

$\Rightarrow m=\pm \sqrt{3}=\tan (\pm \frac{\pi }{3})$