Coordinate Geometry - Parabola

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

$Y^2 = \dfrac{16}{5} X$ $\;\;\; \cdots \; (3)$

  1. 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 $\;$ $\left(\dfrac{19}{4}, 2\right)$

  2. Axis of equation $(3)$ is $\;\;$ $Y = 0$

    $\therefore \;$ Axis of equation $(1)$ is $\;\;\;$ $y - 2 = 0$ $\;\;\;$ [by equation $(2b)$]

    i.e. $\;$ $y = 2$

  3. 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 $\;\;\;$ $\left(\dfrac{111}{20}, 2\right)$

  4. 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 $\;\;\;$ $20 x - 79 = 0$

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