How do you find the equation of the parabola with vertex (0,0) and focus (0,-3)?

Question

4004 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-30T09:27:18

The equation is #y=-1/12x^2#

If the vertex is at #(0,0)# and the focus is at #(0,-3)#

The focus is the mid-point from the vertex to the directrix,

Therefore,

The directrix is #y=3#

Any point #(x,y)# on the parabola is equidistant from the focus and from the directrix.

Therefore,

#(y-3)=sqrt((x-0)^2+(y+3)^2)#

Squaring both sides

#(y-3)^2=(x)^2+(y+3)^2#

#y^2-6y+9=x^2+y^2+6y+9#

#-12y=x^2#

#y=-1/12x^2#

graph{-1/12x^2 [-10, 10, -5, 5]}