What is the equation of the parabola with a focus at (-1,-2) and a directrix of y= -10?

Jan 20, 2016

$y = {x}^{2} / 16 + \frac{x}{8} - \frac{95}{16}$

Explanation:

Let $\left({x}_{0} , {y}_{0}\right)$ be a point on the parabola.
Focus of the parabola is given at $\left(- 1 , - 2\right)$
Distance between the two points is
sqrt((x_0-(-1))^2+(y_0-(-2))^2
or sqrt((x_0+1)^2+(y_0+2)^2
Now distance between the point $\left({x}_{0} , {y}_{0}\right)$ and the given directrix $y = - 10$, is
$| {y}_{0} - \left(- 10\right) |$
$| {y}_{0} + 10 |$

Equate the two distance expressions and squaring both sides.
${\left({x}_{0} + 1\right)}^{2} + {\left({y}_{0} + 2\right)}^{2} = {\left({y}_{0} + 10\right)}^{2}$
or $\left({x}_{0}^{2} + 2 {x}_{0} + 1\right) + \left({y}_{0}^{2} + 4 {y}_{0} + 4\right) = \left({y}_{0}^{2} + 20 {y}_{0} + 100\right)$

Rearranging and taking term containing ${y}_{0}$ to one side
${x}_{0}^{2} + 2 {x}_{0} + 1 + 4 - 100 = 20 {y}_{0} - 4 {y}_{0}$
${y}_{0} = {x}_{0}^{2} / 16 + {x}_{0} / 8 - \frac{95}{16}$

For any point $\left(x , y\right)$ this must be true. Therefore, the equation of the parabola is
$y = {x}^{2} / 16 + \frac{x}{8} - \frac{95}{16}$