Question

Find the standard form of the equation of the parabola with a focus at (0, -10) and a directrix at y = 10

Answer 1

keeping in mind that the distance from the vertex to the focus, is the exact same one as the distance from the vertex to the directrix, in the opposite direction


where do you think is the vertex?

Answer 2

is it 10?

Answer 3

well, if the distance from the vertex to the focus, is the exact same one as the distance from the vertex to the directrix
that means that the vertex is half-way between both of them

Answer 4

So 5

Answer 5

see now where the vertex is?

Answer 6

0?

Answer 7

yeap, is at the origin, 0, 0

notice the distance from the vertex to the focus (0, -10), is 10 units
the "focus form" for a parabola like so is \(\bf (x-h)^2=4p(y-k)\)

(h, k) = vertex coordinates
p = distance from the vertex to the focus point

Answer 8

so i just plug the numbers in? (0 - 0)^2 = 4(10) (-10 - 0)

Answer 9

well, \(\bf (x-h)^2=4p(y-k) \implies (x-0)^2=4(10)(y-0)\)

Answer 10

yes, and then you can solve for "y" or equate to 0

Answer 11

well, it has an "x" and "y", so might as well solve for "y"

Answer 12

x^2 - 40 = y

Answer 13

well, \(\bf \bf (x-h)^2=4p(y-k) \implies (x-0)^2=4(10)(y-0) \implies x^2 = 40y\\
\cfrac{x^2}{40} = y \implies \cfrac{1}{40}x^2 = y \)

Answer 14

well, that's the parabola equation

Answer 15

Oh okay. thank you!

Answer 16

yw