# What is the equation of the parabola with a focus at (2,1) and a directrix of y= 3?

Aug 14, 2017

${x}^{2} - 4 x + 4 y - 4 = 0$

#### Explanation:

$\text{for any point "(x,y)" on the parabola}$

$\text{the distance from "(x,y)" to the focus and directrix are}$
$\text{equal}$

$\text{using the "color(blue)"distance formula}$

$\Rightarrow \sqrt{{\left(x - 2\right)}^{2} + {\left(y - 1\right)}^{2}} = | y - 3 |$

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

${\left(x - 2\right)}^{2} + {\left(y - 1\right)}^{2} = {\left(y - 3\right)}^{2}$

$\Rightarrow {x}^{2} - 4 x + 4 + {y}^{2} - 2 y + 1 = {y}^{2} - 6 y + 9$

$\Rightarrow {x}^{2} - 4 x \cancel{+ {y}^{2}} \cancel{- {y}^{2}} - 2 y + 6 y + 4 + 1 - 9 = 0$

$\Rightarrow {x}^{2} - 4 x + 4 y - 4 = 0 \leftarrow \textcolor{red}{\text{ is the equation}}$