Find the vertex, axis, focus and directrix of the parabola $\;$ $5y^2 - 20y - 16x + 96 = 0$.
Equation of given parabola: $\;$ $5y^2 - 20y - 16x + 96 = 0$
i.e. $\;$ $y^2 - 4y - \dfrac{16}{5}x + \dfrac{96}{5} = 0$
i.e. $\;$ $\left(y^2 - 4y + 4\right) = \dfrac{16}{5} x - \dfrac{96}{5} + 4$
i.e. $\;$ $\left(y - 2\right)^2 = \dfrac{16}{5} x - \dfrac{76}{5}$
i.e. $\;$ $\left(y - 2\right)^2 = \dfrac{16}{5} \left(x - \dfrac{76}{16}\right)$
i.e. $\;$ $\left(y - 2\right)^2 = \dfrac{16}{5} \left(x - \dfrac{19}{4}\right)$ $\;\;\; \cdots \; (1)$
To shift the origin to the point $\left(\dfrac{19}{4}, 2\right)$,
let $\;$ $x - \dfrac{19}{4} = X$ $\;\;\; \cdots \; (2a)$ $\;$ and $\;$ $y - 2 = Y$ $\;\;\; \cdots \; (2b)$
In view of equations $(2a)$ and $(2b)$, equation $(1)$ becomes
$Y^2 = \dfrac{16}{5} X$ $\;\;\; \cdots \; (3)$
- Vertex of equation $(3)$ is $\;$ $\left(0, 0\right)$
i.e. $\;$ $X = 0, \;\;\; Y = 0$
When $\;$ $X = 0$
$\implies$ $x - \dfrac{19}{4} = 0$ $\;$ i.e. $\;$ $x = \dfrac{19}{4}$ $\;\;\;$ [by equation $(2a)$]
When $\;$ $Y = 0$
$\implies$ $y - 2 = 0$ $\;$ i.e. $\;$ $y = 2$ $\;\;\;$ [by equation $(2b)$]
$\therefore \;$ Vertex of equation $(1)$ is $\;$ $\left(\dfrac{19}{4}, 2\right)$
- Axis of equation $(3)$ is $\;\;$ $Y = 0$
$\therefore \;$ Axis of equation $(1)$ is $\;\;\;$ $y - 2 = 0$ $\;\;\;$ [by equation $(2b)$]
i.e. $\;$ $y = 2$
- Comparing equation $(3)$ with the standard equation of parabola $\;\;$ $Y^2 = 4a X$ $\;\;$ gives
$4a = \dfrac{16}{5}$ $\implies$ $a = \dfrac{4}{5}$
$\therefore \;$ Focus of equation $(3)$ is $\;\;\;$ $\left(a, 0\right) = \left(\dfrac{4}{5}, 0\right)$
$\implies$ $X = \dfrac{4}{5}, \;\;\; Y = 0$
When $\;$ $X = \dfrac{4}{5}$
$\implies$ $x - \dfrac{19}{4} = \dfrac{4}{5}$ $\;$ i.e. $\;$ $x = \dfrac{111}{20}$ $\;\;\;$ [by equation $(2a)$]
When $\;$ $Y = 0$
$\implies$ $y - 2 = 0$ $\;$ i.e. $\;$ $y = 2$ $\;\;\;$ [by equation $(2b)$]
$\therefore \;$ Focus of equation $(1)$ is $\;\;\;$ $\left(\dfrac{111}{20}, 2\right)$
- Directrix of equation $(3)$ is $\;\;\;$ $X = -a$
i.e. $\;$ $X = \dfrac{-4}{5}$
i.e. $x - \dfrac{19}{4} = \dfrac{-4}{5}$ $\;\;$ i.e. $\;\;$ $x = \dfrac{79}{20}$ $\;\;\;$ [by equation $(2a)$]
$\therefore \;$ Directrix of equation $(1)$ is $\;\;\;$ $20 x - 79 = 0$