Question

Derive the equation of parabola $y^2=4ax$ in standard form.

Answer 1

As we have parabola

Take a point $P(x,y)$ on the parabola such that, $PF=PB\dots \left(i\right)$

By distance formula,

${\left(PF\right)}^2={(x-a)}^2+y^2$

and ${\left(PB\right)}^2={(x+a)}^2$

As PF = PB [from (i)],

Now, ${(x-a)}^2+y^2={(x+a)}^2$

$\Rightarrow x^2-2ax+a^2+y^2=x^2+2ax+a^2$

$\mathrm{We}\mathrm{get}{\mathrm{y}}^2=4\mathrm{ax}\mathrm{at}[\mathrm{a},0]$ is the standard form.