What is the standard form of the equation of the parabola with a directrix at x=-5 and a focus at (-2,-5)?

Jan 21, 2017

The equation is ${\left(y + 5\right)}^{2} = 6 \left(x + \frac{7}{2}\right)$

Explanation:

Any point $\left(x , y\right)$ on the parabola is equidistant from the directrix and the focus.

Therefore,

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

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

${x}^{2} + 10 x + 25 = {x}^{2} + 4 x + 4 + {\left(y + 5\right)}^{2}$

${\left(y + 5\right)}^{2} = 6 x + 21$

${\left(y + 5\right)}^{2} = 6 \left(x + \frac{7}{2}\right)$

The vertex is $\left(- \frac{7}{2} , - 5\right)$

graph{((y+5)^2-6(x+7/2))(y-100x-500)((x+2)^2+(y+5)^2-0.05)=0 [-28.86, 28.86, -20.2, 8.68]}