Question

The condition for the line $y=mx+c$ to be a normal to the parabola $y=4ax$ is _______

• A. $c=\frac{a}{m}$
• B. $c=2am+a{m}^3$
• C. $c=2am-a{m}^3$
• D. $c=\frac{a}{m}$

Answer 1

Answer: (C)

Given that, equation of parabola $y^2=4ax$, let the parametric coordinate is $\left(a{m}^2,2am\right)$.
$\Rightarrow 2y\frac{dy}{dx}=4a$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}$
Slope of normal $=\left(\frac{-y}{2a}\right)$
At $\left(a{m}^2,2am\right)=\frac{-2am}{2a}=-m$
Now, the equation of normal to the parabola is
$(y-2am)=(-m)\left(x-a{m}^2\right)$
$y-2am=-mx+a{m}^3$
$mx+y-\left(2am+a{m}^3\right)=0$...(i)
Also, given the line
$y=mx+c$ or $mx-y+c=0$...(ii)
is normal to parabola, then
On comparing $c=-2am-a{m}^3$