Derive the equation of the parabola with a focus at (−2, 4) and a directrix of y = 6. Put the equation in standard form.

Question

anonymous · Answer

Derive the equation of the parabola with a focus at (−2, 4) and a directrix of y = 6. Put the equation in standard form.

anonymous · Answer

okay, so the focus is below the directrix, which means this is a parabola that opens downwards and has an equation that has an x².
4p(y - k) = (x - h)² is the general form.

anonymous · Answer

p would be the distance from the directrix to the vertex, or twice the distance from the directrix to the focus.
(h,k) is the co-ords of the vertex.

anonymous · Answer

2|p| = 2 so |p| = 1 (the focus is at y = 4 and the directrix is y = 6). the difference is 2.

anonymous · Answer

since it is facing downwards, p will equal -1.
the vertex is at (-2,5) and the y co-ord is halfway between the focus and the directrix.
the x bit is -2 for the focus and the vertex, so the equation will be -4(y - 5) = (x - -2)² .

anonymous · Answer

so you finish the problem with:
-4(y - 5) = (x + 2)²
-4y +20 = x² +4x + 4
-4y = x² +4x - 16
4y = -x² -4x + 16
and the final equation is:
y = -1/4x² -x + 4

anonymous · Answer

thank u so much. I have one more

anonymous · Answer

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

anonymous · Answer

okay, well you pretty much work this in the same way as you worked the last one. the difference is that this parabola is facing upwards.
you will still use the equation : 4p(y - k) = (x - h)².
this time, 2|p| = 2, which means |p| = 1. (the focus is at y = 1 and the directrix is y = -1) the difference is 2 (1- - 1) .

anonymous · Answer

the vertex is at (0,0) and the y co-ord is halfway between the focus and the directrix. the x bit is 0 for the focus and the vertex, so your equation will be:
4(y - 0) = (x - 0)²
4(y) = (x)²
and the final answer is :
y = 1/4x²

anonymous · Answer

this one was easier than the last one, because the vertex is at the origin (0,0).
that's why there was no x term this time.

anonymous · Answer

thank u so much

anonymous · Answer

Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8.

anonymous · Answer

you're welcome.. but do you not know how to do them at all? xD
try this one on your own. (:

anonymous · Answer

I find  y= 1/8  (x +2)^2 +6

anonymous · Answer

with my calculations, it would actually be y = -1/8 (x - 2)² + 6 .