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+25/8

Answer 1

Distance from (x,y) to focus (3,4) equals distance to directrix y=0 (so y>0):

sqrt((x−3)^2 + (y−4)^2) = y
Square both sides:
(x−3)^2 + (y−4)^2 = y^2
(x−3)^2 − 8y + 16 = 0 → 8y = (x−3)^2 + 16

So
y = ((x−3)^2 + 16)/8 = (x^2 − 6x + 25)/8.

Equivalent forms:
y = 1/8 x^2 − 3/4 x + 25/8
or y = 1/8 (x−3)^2 + 2.