Question

Write the equation for a parabola with a focus at (-8,-1)(?8,?1)left parenthesis, minus, 8, comma, minus, 1, right parenthesis and a directrix at y=-4y=?4y, equals, minus, 4

Answer 1

This will be a parabola which opens upwards. and with a vertex at ( (-8, (-1-1.5) = (-8, -2.5).    (The  vertex is halfway between the focus and the directrix).

The general form is  (x-h)^2 = 4p(y - k)  where (h,k) is the vertex.

So we can write the equation as  

(x  + 8)^2 = 4p(y + 2.5)

p = 1.5  so we have

(x + 8)^2 = 6(y + 2.5)

x^2 + 16x + 64 = 6y + 15

6y = x^2 + 16x + 49
 

y = (1/6) x^2 + (8/3)x + (49/6)  is the answer