Find the equation of the hyperbola if the focus is $\left(2,3\right)$; the corresponding directrix is $x + 2y = 5$ and $e = 2$.
Given: Equation of directrix is $\;$ $x + 2y = 5$; $\;$ $e = 2$
Let $P \left(x, y\right)$ be any point on the hyperbola.
Draw $PM$ perpendicular to the directrix.
By definition, $\dfrac{FP}{PM} = e$
i.e. $\;$ $FP^2 = e^2 \cdot PM^2$
i.e. $\;$ $\left(x - 2\right)^2 + \left(y - 3\right)^2 = \left(2\right)^2 \times \left(\dfrac{x + 2y - 5}{\sqrt{1^2 + 2^2}}\right)^2$
i.e. $\;$ $x^2 - 4x + 4 + y^2 - 6y + 9 = \dfrac{4}{5} \left(x^2 + 4y^2 + 25 + 4xy - 10x - 20y\right)$
i.e. $\;$ $5x^2 - 20x + 5y^2 - 30y + 65 = 4x^2 + 16y^2 + 100 + 16xy - 40x - 80y$
i.e. $\;$ $x^2 - 16xy - 11y^2 + 20x + 50y - 35 = 0$ $\;\;\; \cdots \; (1)$
Equation $(1)$ is the required equation of hyperbola.