Question

Find the equation of the parabola that satisfies the given conditions: focus $(0,-3)$ and directrix $y=3$.

Answer 1

Given: Focus $(0,-3)$ and directrix $y=3$.
Finding $a$
Since focus lies on negative $y-\text{axis}$ and directrix is parallel to $x-\text{axis}$, the $y-\text{axis}$ itself is the axis of the parabola.
Hence the equation of the parabola is of the form either
$x^2=4ay\text{or}{x}^2=-4ay$

Now, focus has negative $y$ co-ordinate
So, we must use equation $x^2=-4ay$
Coordinates of focus $=(0,-a)$
$=(0,-3)$ (Given)
Hence, $a=3$

Equation of Parabola
Required equation is given by:

$\Rightarrow x^2=-4ay$

$\Rightarrow x^2=-4\times 3\times y$

$\Rightarrow x^2=-12y$