Can someone help me with the question: Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8. I got everything up to 1/8 (x-2)^2+6, but I don't know whether or not the 1/8 is supposed to be negative or positive.

Question

owlcoffee · Answer

The parabola is defined as the set of points that are equidistant to a fixed point called the "foci" and a line called "directrix", this of course requires that you use the definition to apply the distance formula.
So, if we let a point P belong to the directrix, line y=8, then we know that any value of x we choose will have it's same y=8 value, so, I will take the point P(1,8) and a general point U(x,y), we know that this point U must be equidistant from the foci and the point P:
$$dist(f,U)=dist(U,P)$$
$$\sqrt{(x-2)^2+(y-4)^2}=\sqrt{(1-x)^2+(8-y)^2}$$
So, simplifying:
$$(x-2)^2+(y-4)^2=(1-x)^2+(8-y)^2$$

anonymous · Answer

Yes, @Owlcoffee , but that is where I am stuck...  simplifying. Would you mind helping a little further?