Question

Derive the equation of a parabola given the focus is at (3,4)
(
3
,
4
)
and the directrix is y=0
𝑦
=
0
. Fill in the missing values of the equation in standard form.(1 point)
y=
𝑦
=
x2−34x+258

Answer 1

Distance from (x,y) to focus = distance to directrix:
sqrt((x-3)^2 + (y-4)^2) = y.

Square and simplify:
(x-3)^2 + (y-4)^2 = y^2
x^2 - 6x + 9 + y^2 - 8y + 16 = y^2
x^2 - 6x + 25 - 8y = 0
8y = x^2 - 6x + 25

So the parabola is
y = (1/8)x^2 - (3/4)x + 25/8,

equivalently (vertex form)
(x - 3)^2 = 8(y - 2).