Question

From a point $(C,0)$ three normals are drawn to the parabola $y^2=x$. Then,

• A. $C<\frac{1}{2}$
• B. $C=\frac{1}{2}$
• C. $C>\frac{1}{2}$
• D. $\frac{1}{2}>C>\frac{1}{4}$

Answer 1

Answer: (C)

Given, $y^2=x$
$\therefore 4a=1$
$\Rightarrow a=\frac{1}{4}$
Any normal to the parabola is
$y=mx-2am-a{m}^3...(i)$
If it passes through $(C,0)$, then
$0=mC-\frac{1}{2}m-\frac{m^3}{4}\left\{\because a=\frac{1}{4}\right\}$
$\Rightarrow m=0$ and $\frac{m^2}{4}=C-\frac{1}{2}$
$\therefore C>\frac{1}{2}$