Question

The coordinates of the points on the parabola $y^2=8x$ whose focal distance is 4 are $\underset{–}{}$

Answer 1

Given : Equation of parabola is $y^2=8x$
$\Rightarrow y^2=4\times 2\times x$
$\Rightarrow a=2$
$\therefore \text{Focus, S=(2, 0)}$
Let a point be $P(2t^2,4t)$ on the parabola
Given that- Focal distance, $SP=4$
$\Rightarrow S{P}^2=4^2$
$\Rightarrow (2t^2-2{)}^2+(4t-0{)}^2=16$
$\Rightarrow 4t^4-8t^2+4+16t^2=16$
$\Rightarrow t^4+2t^2+1=4$
$\Rightarrow (t^2+1{)}^2=2^2$
$\Rightarrow t^2+1=2$
$\Rightarrow t^2=1$
$\Rightarrow t=\pm 1$
When $t=1$$\Rightarrow P=(2,4)$
When $t=-1$$\Rightarrow P=(2,-4)$

Hence, the coordinates of points on parabola $y^2=8x$ whose focal distance is $4$ are $(2,4)$ and $(2,-4)$.