Question

The points $\left(0,4\right)$ and $\left(0,2\right)$ are respectively the vertex and focus of a parabola. Then find the equation of the parabola.

Answer 1

The coordinates of the vertex is (0,4)

The coordinates of the focus is (0,2)

It is clear that the vertex and the focus lies on the positive side of the y-axis.

Hence the curve is open downwards.

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

i.e. $(x-0{)}^2=-4\times 2(y-4)$

On simplifying we get,

$x^2=-8(y-4)$

$\Rightarrow x^2=-8y+32$

$x^2+8y-32=0$ is the required equation of the parabola.