What is the equation in standard form of the parabola with a focus at (1,4) and a directrix of y= 2?

1 Answer
Dec 6, 2015

y=1/4x^2-1/2x+13/4

Explanation:

If (x,y) is a point on a parabola then
color(white)("XXX")the perpendicular distance from the directrix to (x,y)
is equal to
color(white)("XXX")the distance from (x,y) to the focus.

If the directrix is y=2
then
color(white)("XXX")the perpendicular distance from the directrix to (x,y) is abs(y-2)

If the focus is (1,4)
then
color(white)("XXX")the distance from (x,y) to the focus is sqrt((x-1)^2+(y-4)^2)
enter image source here
Therefore
color(white)("XXX")color(green)(abs(y-2)) = sqrt(color(blue)((x-1)^2)+color(red)((y-4)^2))

color(white)("XXX")color(green)(y-2)^2) = color(blue)((x-1)^2)+color(red)((y-4)^2)

color(white)("XXX")color(green)(cancel(y^2)-4y+4) = color(blue)(x^2-2x+1) + color(red)(cancel(y^2)-8y+16)

color(white)("XXX")4y + 4 = x^2-2x+17

color(white)("XXX")4y = x^2 -2x +13

color(white)("XXX")y = 1/4x^2 -1/2x + 13/4color(white)("XXX")(standard form)
graph{1/4x^2-1/2x+13/4 [-5.716, 6.77, 0.504, 6.744]}