Question

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

y=14x2+x+4
𝑦
=
1
4
𝑥
2
+
𝑥
+
4

y equals Start Fraction 1 over 4 End Fraction x squared plus x plus 4

y=−14x2+3x−10
𝑦
=
−
1
4
𝑥
2
+
3
𝑥
−
10

y equals negative Start Fraction 1 over 4 End Fraction x squared plus 3 x minus 10

y=−14x2+2x−6
𝑦
=
−
1
4
𝑥
2
+
2
𝑥
−
6

y equals negative Start Fraction 1 over 4 End Fraction x squared plus 2 x minus 6

y=14x2+3x+8
𝑦
=
1
4
𝑥
2
+
3
𝑥
+
8

y equals Start Fraction 1 over 4 End Fraction x squared plus 3 x plus 8

Answer 1

Use the distance definition: distance from (x,y) to focus (6,-2) equals distance to the line y=0.

sqrt[(x−6)^2 + (y+2)^2] = |y|. Square both sides:
(x−6)^2 + (y+2)^2 = y^2
(x−6)^2 + 4y + 4 = 0
4y = −(x−6)^2 − 4
y = −1 − 1/4 (x−6)^2

Expand:
y = −(1/4)x^2 + 3x − 10.

So the correct choice is y = −(1/4)x^2 + 3x − 10.