Find the equation of the ellipse if one of the foci is $\left(0,-1\right)$, the corresponding directrix is $\;$ $3x + 16 = 0$ $\;$ and $\;$ $e = \dfrac{3}{5}$
Let $\;$ $P \left(x,y\right)$ $\;$ be a moving point.
Focus $= F = \left(0,-1\right)$
Equation of directrix is $\;$ $3x + 16 = 0$
Draw $\;$ $PM$ $\;$ perpendicular to the directrix.
By definition, $\;$ $\dfrac{FP}{PM}= e$
$\therefore$ $\;$ $FP^2 = e^2 \; PM^2$
i.e. $\;$ $\left(x - 0\right)^2 + \left(y + 1\right)^2 = \left(\dfrac{3}{5}\right)^2 \times \left(\dfrac{\left|3x + 16\right|}{\sqrt{3^2}}\right)^2$
i.e. $\;$ $x^2 + y^2 + 2y + 1 = \dfrac{9}{25} \times \left(\dfrac{9x^2 + 96x + 256}{9}\right)$
i.e. $\;$ $25x^2 + 25y^2 + 50y + 25 = 9x^2 + 96x + 256$
i.e. $\;$ $16x^2 + 25y^2 - 96x + 50y - 231 = 0$ $\;\;\; \cdots \; (1)$
Equation $(1)$ is the equation of the required ellipse.