Question

Find the equation of the parabolas whose vertex is $at(0,4)$ and focus is $at(0,2)$ ?

Answer 1

From the given detail is clear that the parabola is open downwards.

Hence the equation is of the form${(x-h)}^2=-4a(y-k)$

Given vertex V is $(0,4)$ and focus $(0,2)$.

The distance between the vertex and the focus is $VF=a$

$VF=\sqrt{{(0-0)}^2+{(2-4)}^2}=2$

Hence $a=2$

∴The required equation of the parabola is $(x-0{)}^2=-4(2\left)\right(y-4)$

$x^2=-8(y-4)$ = $x^2=32-8y$ is the required equation of the parabola.