Question

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

Answer 1

Comparing above parabola with $y^2=4ax$, we get $a=\frac{1}{2}$

So focal distance is given by, $x+a=\frac{5}{2}$ (given )

$\Rightarrow x+\frac{1}{2}=\frac{5}{2}\Rightarrow x=2$

Substitute $x=2$ in the given parabola,

$y^2=2x=2^2\Rightarrow y=\pm 2$

Hence coordinate of the points on the parabola whose focal distance is $\frac{5}{2}$ units are $(2,2)$ and $(2,-2)$