Q. Three distinct normals to the parabola $y^{2} = x$ are drawn through a point $(c,\ 0)$, then
A) $c = \frac{1}{4}$
B) $c = \frac{1}{2}$
C) $c > \frac{1}{2}$
D) None of these
Explanation
(c)
Equation of normal is $y = mx - \frac{m}{2} - \frac{m^{3}}{4}\left( a = \frac{1}{4} \right).$ It passes through $(c,\ 0)$
$\therefore 0 = cm - \frac{m}{2} - \frac{m^{3}}{2} \Rightarrow m = 0$
And $\frac{m^{2}}{4} = c - \frac{1}{2} \Rightarrow c > \frac{1}{2}$
Then, all values of $m$ are real
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.