Question

Find the co-ordinates of the points on the parabola $y^2=2x$ whose focal distance is $\frac{5}{2}$

Answer 1

A point on the parabola $y^2=2x$ can be written as $(2t^2,2t)$

The focus of $y^2=2x$ is $(\frac{1}{2},0)$

$\therefore \frac{5}{2}=\sqrt{{(2t^2-\frac{1}{2})}^2+4t^2}$

$\therefore \frac{25}{4}=4t^4+\frac{1}{4}+4t^2-2t^2$

$\therefore \frac{25}{4}=4t^4+\frac{1}{4}+2t^2$

$t^2+\frac{1}{2}=\pm \frac{5}{2}$

$\therefore t^2=2$ or $t^2=-3$

Since $t^2$ cannot be negative, the points are $(4,2\sqrt{2})$ and $(4,-2\sqrt{2})$