Question

Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

Answer 1

the general form is

\[(x - h)^2 = 4a(y - k)\]

the distance from the focus to the directrix is 2a

and the vertex is halfway between the directrix and focus on the line of symmetry x = 0

sin the focus is (0, -4) the line of symmetry is x = 0

so 2a = 8

a = 4

the vertex is 4 above the focus as the parabola is concave down

graph the information

so find the vertex and the equation is

\[(x - h)^2 = -4 \times 4 \times (y - k)\]

where (h, k) is the vertex.

hope it helps