Question

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

y=−14x2+3x−10
y equals negative Start Fraction 1 over 4 End Fraction x squared plus 3 x minus 10

y=−14x2+2x−6
y equals negative Start Fraction 1 over 4 End Fraction x squared plus 2 x minus 6

y=14x2+x+4
y equals Start Fraction 1 over 4 End Fraction x squared plus x plus 4

y=14x2+3x+8

Answer 1

Since the directrix is at y=0, the vertex of the parabola is at (6, 0) which is half-way between the focus and directrix. The distance between the focus and vertex is equal to the distance between the directrix and the vertex, with a length of 2a (where a is the distance from the vertex to the focus or directrix).

Since the focus is at (6, -2), the focus is a distance of a = 2 units below the vertex.

Therefore, the equation of the parabola is: y = -1/4(x-6)^2 - 2

Expanding that equation gives: y = -1/4(x^2 - 12x + 36) - 2
y = -1/4x^2 + 3x - 9 - 2
y = -1/4x^2 + 3x - 11

So, the correct equation is: y = -1/4x^2 + 3x - 11