# What is the vertex form of the equation of the parabola with a focus at (12,22) and a directrix of y=11 ?

Nov 24, 2017

$y = \frac{1}{22} {\left(x - 12\right)}^{2} + \frac{33}{2}$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$

$\text{for any point "(x.y)" on a parabola}$

$\text{the focus and directrix are equidistant from } \left(x , y\right)$

$\text{using the "color(blue)"distance formula "" on "(x,y)" and } \left(12 , 22\right)$

$\Rightarrow \sqrt{{\left(x - 12\right)}^{2} + {\left(y - 22\right)}^{2}} = | y - 11 |$

$\textcolor{b l u e}{\text{squaring both sides}}$

$\Rightarrow {\left(x - 12\right)}^{2} + {\left(y - 22\right)}^{2} = {\left(y - 11\right)}^{2}$

${\left(x - 12\right)}^{2} \cancel{+ {y}^{2}} - 44 y + 484 = \cancel{{y}^{2}} - 22 y + 121$

$\Rightarrow {\left(x - 12\right)}^{2} = 22 y - 363$

$\Rightarrow y = \frac{1}{22} {\left(x - 12\right)}^{2} + \frac{33}{2} \leftarrow \textcolor{red}{\text{in vertex form}}$