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)
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]}