Question

Prove that the equation to the parabola, whose vertex and focus are on the axis of $x$ at distances $a$ and $a^{\prime }$ from the origin respectively, is $y^2=4(a^{\prime }-a)(x-a).$

Answer 1

The general equation of parabola is ${(y-k)}^2=4A(x-h)$ where $(h,k)$ is the vertex of of the parabola

$A$ is the distance between the vertex and the focus, so in our case

$A=a^{\prime }-a$

And vertex is $(a,0)$

So, the equation of parabola is

$y^2=4(a^{\prime }-a)(x-a)$