Find the equation of the parabola if the focus is $\left(2, -3\right)$ and the directrix is $2y - 3 = 0$.
Let $P \left(x,y\right)$ be any point on the parabola with focus $F \left(2, -3\right)$ and directrix $\;$ $2y - 3 = 0$.
Draw $PM$ perpendicular to the directrix.
For a parabola, $\dfrac{FP}{PM} = \text{eccentricity} = e = 1$
$\implies$ $FP^2 = PM^2$ $\;\;\; \cdots \; (1)$
Now, $FP^2 = \left(x - 2\right)^2 + \left(y + 3\right)^2$ $\;\;\; \cdots \; (2a)$
and $PM^2 = \left(\pm \dfrac{2y - 3}{\sqrt{2^2}}\right) = \dfrac{\left(2y - 3\right)^2}{4}$ $\;\;\; \cdots \; (2b)$
$\therefore$ $\;$ In view of equations $(2a)$ and $(2b)$ equation $(1)$ becomes
$\left(x - 2\right)^2 + \left(y + 3\right)^2 = \dfrac{\left(2y - 3\right)^2}{4}$
i.e. $x^2 - 4x + 4 + y^2 + 6y + 9 = \dfrac{4y^2 - 12y + 9}{4}$
i.e. $4x^2 - 16 x + 4 y^2 + 24 y + 52 = 4y^2 - 12y + 9$
i.e. $4x^2 - 16x + 36 y + 43 = 0$ $\;\;\; \cdots \; (3)$
Equation $(3)$ is the equation of the required parabola.