Find the standard form of the equation of the parabola with a focus at (0, 4) and a directrix at y = -4.

Answer 1

Focus of parabola (0,4) and directrix y  = -4

If a parabola has a horizontal axis, the standard form of the parabola is (x - h )² = 4p (y - k ).

Where p not equals to 0.Vertex (h,k ) ,focus (h , k+p ) and directrix y  = k - p

(h ,k+p ) = (0,4)

h  = 0

k + p  = 4     ---> (1)

Directrix y  = k - p = - 4

k - p = - 4      ---> (2)

Add equations (1) and (2):

2k=0

k  = 0

Substitute 0 for k in equation (1).

k+p  = 4

0+p=4

p=4

So the vertex is (0 , 0)

Substitute the values of h ,k and p in standard form of parabola.

Equation of parabola is x² = 16y.