Q. The chord $AB$ of the parabola $y^{2} = 4ax$ cuts the axis of the parabola at $C$. If $A = \left( at_{1}^{2},\ 2at_{1} \right),\ B = (at_{2}^{2},\ 2at_{2})$ and $AC:AB = 1:3,$ then
A) $t_{2} = 2t_{1}$
B) $t_{2} + {2t}_{1} = 0$
C) $t_{1} + 2t_{2} = 0$
D) None of these
Explanation
(b)
$\because A\left( at_{1}^{2},\ 2at_{1} \right)$ and $B(at_{2}^{2},\ 2at_{2})$ are such that $AC:AB = 1:3$
$\therefore$ Coordinates of $C$ are $\left( \frac{2at_{1}^{2} + at_{2}^{2}}{3},\ \frac{4at_{1} + 2at_{2}}{3} \right)$
Point $C$ lies on $x$-axis, then
$\frac{4at_{1} + 2at_{2}}{3} = 0$
$\Rightarrow t_{2} + 2t_{1} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.