# What is the standard form of the equation of the parabola with a directrix at x=-5 and a focus at (-7,-5)?

Oct 31, 2017

The equation of the parabola is ${\left(y + 5\right)}^{2} = - 4 x - 24 = - 4 \left(x + 6\right)$

#### Explanation:

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

Therefore,

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

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

Squaring and developing the ${\left(x + 7\right)}^{2}$ term and the LHS

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

${x}^{2} + 10 x + 25 = {x}^{2} + 14 x + 49 + {\left(y + 5\right)}^{2}$

${\left(y + 5\right)}^{2} = - 4 x - 24 = - 4 \left(x + 6\right)$

The equation of the parabola is ${\left(y + 5\right)}^{2} = - 4 x - 24 = - 4 \left(x + 6\right)$

graph{((y+5)^2+4x+24)((x+7)^2+(y+5)^2-0.03)(y-100(x+5))=0 [-17.68, 4.83, -9.325, 1.925]}