Question

I actually have 5 questions on the same thing, just different setups. But how would I go about solving them? .-.

1. Derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7
2. Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1
3. Derive the equation of the parabola with a focus at (−5, 5) and a directrix of y = -1.
4. Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4
5. Derive the equation of the parabola with a focus at (−2, 4) and a directrix of y = 6. Put the equation in standard form.

Answer 1

Answer: For 1: f(x)=-1/24(x+5)^2+1

Answer 2

1) 2p = -2.
4p [ y - k ] = [ x - h) ]² --- > - 4 [ y + 5 ] = [ x + 5 ]²
2) 4p * (y - k) = (x - h)^2

(h , k) is the vertex

The vertex is halfway between the focus and the directrix (when they're at their closest)
p is that distance

2 - 1 = 1
4p = 1
p = 1/4

(1/4) * (y - k) = (x - h)^2
y - k = 4 * (x - h)^2

The vertex is at (6 , 3/2), since that's midway between (6 , 1) and (6 , 2)

y - 3/2 = 4 * (x - 6)^2
y = (3/2) + 4 * (x - 6)^2

4) f(x) = (-1/16)*(x²)

5) f(x) = −1/4 x2 − x + 5