Question

If three normals are drawn from the point $\left(c , 0\right)$ to the parabola $y^{2}=4x$ and two of which are perpendicular, then the value of $c$ is equal to

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (A)

Equation of the normal in slope form is $y=mx-2m-m^{3}$ which passes through $\left(c , 0\right)$
$\Rightarrow 0=mc-2m-m^{3}$
$\Rightarrow m^{3}+\left(2 - c\right)m=0$
$\Rightarrow m=0$ or $m^{2}+\left(2 - c\right)=0$
$\Rightarrow m_{1}m_{2}=\frac{2 - c}{1}=-1$
$\Rightarrow c=3$