P and Q are two distinct points on the parabola y^2=4x with parameters t and t_1 respectiv…

Question

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

Accepted Answer

For $y^2=4ax$ with $a=1$, if the normal at parameter $t$ meets the curve again at parameter $t_1$, then $t_1=-t-\dfrac{2}{t}$.

$$t_1^2=t^2+\frac{4}{t^2}+4\ \ge\ 2\sqrt{t^2\cdot\frac{4}{t^2}}+4=4+4=8,$$

by AM--GM, with equality when $t^2=2$. So the minimum value of $t_1^2$ is $8$.

Solutions · 1