Question

The line $lx+my+n=0$ is a normal to the parabola $y^2=4ax$ if

• A. $al(l^2+2m^2)+m^{2}n=0$
• B. $al(l^2+2m^2)=m^{2}n$
• C. $al(2l^2+m^2)=-m^{2}n$
• D. $al(2l^2+m^2)=2m^{2}n$

Answer 1

The correct option is C $al(2l^2+m^2)=-m^{2}n$

Given Parabola: $y^2=4ax...............\left(1\right)$

Normal: $lx+my+n=0$

$=>y=-\frac{l}{m}x-\frac{n}{m}.................\left(2\right)$

Now condition of normal, $C=-2aM-a{M}^3$ ($M=$ slope of Normal, $C=$ constant)

$=>-\frac{n}{m}=-2a(-\frac{l}{m})-a({\frac{-l}{m})}^3$

$=>-m^{2}n=2al{m}^2+a{l}^3$

$=>al(2m^2+l^2)=-m^{2}n.$