Question

Derive the equation of a parabola with a focus at (6,−2) and a directrix at y=0 .(1 point)

y=14x2+3x+8

y=−14x2+2x−6

y=−14x2+3x−10

y=14x2+x+4

Answer 1

Distance to focus = distance to directrix:
sqrt((x−6)^2+(y+2)^2) = |y|. Square both sides:
(x−6)^2 + (y+2)^2 = y^2
x^2 −12x +36 + y^2 +4y +4 = y^2
x^2 −12x +40 +4y = 0
4y = −x^2 +12x −40
y = −(1/4)x^2 +3x −10.

This matches the choice written as y = −14x2+3x−10 (interpreting 14x2 as 1/4 x^2), so select that option.