Question

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

Answer 1

Standard form: `( x + 3 )^2 = -10 ( y -1/2)`
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 `(-3, 1/2)`,

The size parameter a = VS = `5/2`.

Now, the equation is as given in the answer,

Note that 4a =`4 X 5/2` = 10 and the negative sign is prefixed for the axis being in the negative direction of y-axis..