Question

Let $P$ and $Q$ be two distinct points on the parabola $y^2=4x$, with parameters $t$ and $t_1$ respectively. If the normals at $P$ passes through $Q,$ then the minimum value of $t_1^2$ is

Answer 1

Normal at $P\left(t\right)$ passes through $Q\left(t_1\right)$, then
$t_1=-t-\frac{2}{t}$
$\Rightarrow t^2+t{t}_1+2=0$
Since $t$ is real so,
$b^2-4ac\ge 0$
$\Rightarrow t_1^2-8\ge 0\Rightarrow t_1^2\ge 8$

Hence, the minimum value of $t_1^2$ is $8$