math (1209) - Locus: parabolas

apparently, if you make a point (focus, let it = S) within a parabola, and the distance between that point (let it = a) is the distance between the parabola and a line (x = + or - a as the equation depending) then.. the distance from that focus point, to a point on the parabola (P) is equal to the distance from that point, down to the point on the x = a line that shares the same x-coordinate as P, and that would be B.

confusing?

diagrams help.

basically.

PS = PB

use distance formula, get rid of square root

we get:

(x - 0)^2 + (y - a)^2 = (x - x)^2 + (y - -a)^2

x^2 + y^2 - 2ay + a^2 = y^2 +2ay + a^2

cancel out the crap..

x^2 = 4ay

just saying, this was a parabola with vertex at the origin.

to change this, just sub in the different coordinates into the distance formula.

likewise..

if the formula was eg

-x^2 = 4ay

it would be concave down, coz rearranging to make y the subject would make the x^2 negative