What Is the General Equation of a Parabola with Directrix x=p and Focus (h, k)?

[you_of_eh]

For a parabola whose Directrix is given by the equation x=p and whose Focus is (h,k).

Is this by any chance the correct general form of the parabola?

x=1/2(h-p) [y^2 - 2yk + h^2+k^2-p^2]

 

[HallsofIvy]

For a parabola whose Directrix is given by the equation x=p and whose Focus is (h,k).

Is this by any chance the correct general form of the parabola?

x=1/2(h-p) [y^2 - 2yk + h^2+k^2-p^2]

The vertex of a parabola is halfway between the focus and directrix. Here, that is at ((h+p)/2, k) so the focal length is (h+p)/2- p= (h-p)/2. Since the "standard" parabola, with horizontal axis, is 4d(x- x_0)= (y- y_0)^2, here that would be [4(h-p)/2](x- (h+p)/2)= (y- k)^2 which can be written as x= \frac{1}{2(h-p)}(y- k)^2+ \frac{h+p}{2}. If you take that "(h+p)/2" inside the parentheses you get exactly what you have. Well done!

 

[you_of_eh]

Alright perfect! thanks a lot for your time.