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

Answer: (A)

A.P.

Normal to the parabola $y^2=4ax$ with slope $m$ is given by,

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

Given it passes through $(4a,0)$

$\Rightarrow 0=4am-2am-a{m}^3$

$\Rightarrow m^3-2m=0$

$\Rightarrow m(m+\sqrt{2})(m-\sqrt{2})=0$

$\Rightarrow m=-\sqrt{2},0,\sqrt{2}$

Clearly they are in A.P