What is the equation of the parabola with a focus at (-1,-4) and a directrix of y= -7?

Jul 26, 2016

$6 y = {x}^{2} + 2 x - 32$.

Explanation:

Let the Focus be $S \left(- 1 , - 4\right)$ and, let the Directrix be $d : y + 7 = 0$.

By the Focus-Directrix Property of Parabola, we know that, for any pt. $P \left(x , y\right)$ on the Parabola,

$S P = \bot$ Distance $D$ from P to line $d$.

$\therefore S {P}^{2} = {D}^{2}$.

$\therefore {\left(x + 1\right)}^{2} + {\left(y + 4\right)}^{2} = | y + 7 {|}^{2}$

$\therefore {x}^{2} + 2 x + 1 = {\left(y + 7\right)}^{2} - {\left(y + 4\right)}^{2}$

$= \left(y + 7 + y + 4\right) \left(y + 7 - y - 4\right) = \left(2 y + 11\right) \left(3\right) = 6 y + 33$

Hence, the Eqn. of the Parabola is given by,

$6 y = {x}^{2} + 2 x - 32$.

Recall that the formula to find the $\bot$ distance from a pt.$\left(h , k\right)$ to a line $a x + b y + c = 0$ is given by $| a h + b k + c \frac{|}{\sqrt{{a}^{2} + {b}^{2}}}$.