Question

The slopes of the normals to the parabola $y^2=4ax$ intersecting at a point on the axis of the parabola at a distance $4a$, from its vertex are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A A.P.

Equation for normal to the parabola $y^2=4ax$ is
$\begin{array}{}y=mx-2am-a{m}^3......& \text{(1)}\end{array}$
Normal intersecting at a point on the axis of the parabola at a distance $4a$ from its vertex i.e at $(4a,0)$
Substitute the point in equation.... $\left(1\right)$
$\Rightarrow 0=4am-2am-a{m}^3\Rightarrow am(m^2-2)=0$
$\Rightarrow m=0,\pm \sqrt{2}$
Therefore roots can be written as $-\sqrt{2},0,\sqrt{2}$ which is in A.P.