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

Nov 12, 2017

The equation of the parabola is ${\left(x + 1\right)}^{2} = 3 \left(y - \frac{5}{4}\right)$

Explanation:

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

Therefore,

$y - \left(\frac{1}{2}\right) = \sqrt{{\left(x - \left(- 1\right)\right)}^{2} + {\left(y - \left(2\right)\right)}^{2}}$

$y - \frac{1}{2} = \sqrt{{\left(x + 1\right)}^{2} + {\left(y - 2\right)}^{2}}$

Squaring and developing the ${\left(y - 2\right)}^{2}$ term and the LHS

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

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

${\left(x + 1\right)}^{2} = 3 y - \frac{15}{4} = 3 \left(y - \frac{5}{4}\right)$

The equation of the parabola is ${\left(x + 1\right)}^{2} = 3 \left(y - \frac{5}{4}\right)$

graph{((x+1)^2-3(y-5/4))(y-1/2)((x+1)^2+(y-2)^2-0.04)=0 [-12.66, 12.65, -6.33, 6.33]}