Q. The normal at the point ${(bt}_{1}^{2},2{bt}_{1})$ on a parabola $y^{2} = 4bv$ meets the parabola again int he point ${(bt}_{2}^{2},\ {2bt}_{2})$, then
A) $t_{2} = - t_{1} - \frac{2}{t_{1}}$
B) $t_{2} = - t_{1} - \frac{2}{t_{1}}$
C) $t_{2} = t_{1} - \frac{2}{t_{1}}$
D) $t_{2} = t_{1} + \frac{2}{t_{1}}$
Explanation
(a)
Equation of the normal at point $\left( bt_{1}^{2},\ 2{bt}_{1} \right)$ on parabola is
$y = - t_{1}x + 2{bt}_{1} + {bt}_{1}^{3}$
It is also passes through ${(bt}_{2}^{2},{2bt}_{2})$, then
$2bt_{2} = t_{1}.{bt}_{2}^{2} + 2{bt}_{1} + bt_{1}^{3}$
$\Rightarrow \ 2t_{2} - {2t}_{1} = t_{1}\left( t_{1}^{2} - t_{1}^{2} \right)$
$\Rightarrow \ \ \ \ 2 = - t_{1}(t_{2} + t_{1})$
$\Rightarrow \ t_{2} = - t_{1} - \frac{2}{t_{1}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.