# What is the equation of the parabola with a focus at (8,2) and a directrix of y= 5?

Oct 26, 2017

The equation is ${\left(x - 8\right)}^{2} = - 3 \left(2 y - 7\right)$

#### Explanation:

Any point on the parabola is equidistant from the focus and the directrix

Therefore,

$\sqrt{\left(x - 8\right) + \left(y - 2\right)} = 5 - y$

Squaring,

${\left(x - 8\right)}^{2} + {\left(y - 2\right)}^{2} = {\left(5 - y\right)}^{2}$

${\left(x - 8\right)}^{2} + {\cancel{y}}^{2} - 4 y + 4 = 25 - 10 y + {\cancel{y}}^{2}$

${\left(x - 8\right)}^{2} = - 6 y + 21$

${\left(x - 8\right)}^{2} = - 3 \left(2 y - 7\right)$

graph{((x-8)^2+3(2y-7))(y-5)((x-8)^2+(y-2)^2-0.1)=0 [-32.47, 32.47, -16.24, 16.25]}

Oct 26, 2017

${x}^{2} - 16 x + 6 y + 43 = 0$

#### Explanation:

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

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

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

$\Rightarrow \sqrt{{\left(x - 8\right)}^{2} + {\left(y - 2\right)}^{2}} = | y - 5 |$

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

${\left(x - 8\right)}^{2} + {\left(y - 2\right)}^{2} = {\left(y - 5\right)}^{2}$

$\Rightarrow {x}^{2} - 16 x + 64 + {y}^{2} - 4 y + 4 = {y}^{2} - 10 y + 25$

$\Rightarrow {x}^{2} - 16 x + 64 \cancel{+ {y}^{2}} - 4 y + 4 \cancel{- {y}^{2}} + 10 y - 25 = 0$

$\Rightarrow {x}^{2} - 16 x + 6 y + 43 = 0$