Normal at t1 meets the parabola again at t2 - Practice Questions & MCQ

Normal at t₁ meets the parabola again at t₂

Equation of Normal at $\mathrm{P} \equiv\left(\mathrm{at}_1^2, 2 \mathrm{at}_1\right)$ to the parabola $\mathrm{y}^2=4 \mathrm{ax}$ is

$
\mathrm{y}=-\mathrm{t}_{1 \mathrm{x}}+2 \mathrm{at}_1+\mathrm{at}_1^3
$
It meets the parabola again at $\mathrm{Q} \equiv\left(\mathrm{at}_2^2, 2 \mathrm{at}_2\right)$

$
\therefore 2 \mathrm{at}_2=-\mathrm{at}_1 \mathrm{t}_2^2+2 \mathrm{at}_1+\mathrm{at}_1^3
$
$
\begin{aligned}
& \Rightarrow 2 a\left(t_2-t_1\right)+a t_1\left(t_2^2-t_1^2\right)=0 \\
& \Rightarrow \mathrm{a}\left(\mathrm{t}_2-\mathrm{t}_1\right)\left[2+\mathrm{t}_1\left(\mathrm{t}_2+\mathrm{t}_1\right)\right]=0 \\
& \therefore 2+\mathrm{t}_1\left(\mathrm{t}_1+\mathrm{t}_2\right)=0
\end{aligned}
$
$
\mathrm{t}_2=-\mathrm{t}_1-\frac{2}{\mathrm{t}_1}
$