# How do you find the equation of the parabola with directrix y=3 and focus (-3,-2)?

Mar 13, 2016

Standard form: ${\left(x + 3\right)}^{2} = - 10 \left(y - \frac{1}{2}\right)$
Rearranging, ${x}^{2} + 6 x + 10 y + 4 = 0$.

#### Explanation:

The axis of the parabola is perpendicular to the directrix through the focus S. So, its equation is x = $-$3, in the negative direction of y-axis..

The vertex V is on the axis, midway in-between focus S and directrix.
So, V is at $\left(- 3 , \frac{1}{2}\right)$,

The size parameter a = VS = $\frac{5}{2}$.

Now, the equation is as given in the answer,

Note that 4a =$4 X \frac{5}{2}$ = 10 and the negative sign is prefixed for the axis being in the negative direction of y-axis..