write the equation of a parabola with focus (0,4) and directrix y=-4
Some books give the standard form of a y-parabola as
(1) y = a(x - h)² + k
and others give this as its standard form.
(2) (x - h)² = 4p(y - k)
It doesn't matter which your book uses. They are equivalent if a = 1/(4p)
The vertex of a parabola is the point halfway between the focus and the
directrix. The point halfway between the focus (0,4) and the horizontal
directrix y=-4 is the origin (0,0), so if your book uses (1), then it is of
the form
(3) y = ax²
If your book uses (2), then it is of the form:
(4) x² = 4py
p represents the distance from the vertex to the focus, which is considered a
positive number if the directrix is below the vertex and focus, and negative
number if the directrix is above the vertex and focus.
In this case p = +4, so a = 1/(4(4)) or 1/16
If your book uses form (1), then the answer is
(5) y = (1/16)x²
If your book uses form (2), then the answer is x² = 4(4)y or
(6) x² = 16y
Edwin
AnlytcPhil@aol.com
