Question

What is the equation in standard form of the parabola with a focus at (-14,-1) and a directrix of y= -32?

Answer 1

`62y = x^2 +28x - 860`

Explanation:

The equation of the parabola can be written as `4p(y- k)= (x-h)^2` from which the vertex is `(h,k)` and `p` is the distance from the vertex to the focus (or the directrix since the directrix is an equal distance the other side of the vertex).

The focus and directrix are `(-1 - (-32))` units apart and p is therefore `1/2(31) = 31/2`
The vertex is at `(-14,-1-31/2) = (-14,-33/2)`
The equation is therefore `4*31/2 (y+33/2) = (x+14)^2`
`62y +32*33 = x^2 +28x +14^2`
`62y = x^2 +28x +196 - 1056`
`62y = x^2 +28x - 860`