Question

Let F=(0,0) be the focus and the line y=-6 be the directrix. Plot several points P that are half as far from the focus as they are from the directrix. The configuration of all such points is an ellipse. Identify the four points where the ellipse crosses the coordinate axes (two on each axis). Use the distance formula to write an equation for the ellipse.

How I tried solving this: I created two distance equations between a point (x,y) and the focus and the directrix, which I just made into the point (0,-6). When I set those two equations equal to each other while multiplying the equation with the directrix by 1/2, I got x^2+y^2=x+12 which doesn't seem right.

Answer 1

you didn't show your work, but I get for any point (x,y) on the curve:
√(x^2+y^2) = 1/2 (y+6)
x^2+y^2 = 1/4 (y^2+12y+36)
4x^2 + 4y^2 = y^2 + 12y + 36
4x^2 + 3y^2 - 12y = 36
4x^2 + 3(y^2-4y+4) = 36+3*4
4x^2 + 3(y-2)^2 = 48

x^2/12 + (y-2)^2/16 = 1