The general equation of a parabola with a directrix defined by x=p and a focus at (h, k) is accurately represented by the formula x=1/2(h-p) [y^2 - 2yk + h^2 + k^2 - p^2]. The vertex of this parabola is located at ((h+p)/2, k), and the focal length is calculated as (h-p)/2. The standard form of a horizontally oriented parabola is expressed as 4d(x - x_0) = (y - y_0)^2, which can be transformed into the derived equation. This confirms the correctness of the provided formula.
PREREQUISITESMathematicians, physics students, educators, and anyone interested in the geometric properties of parabolas and their applications in various fields.
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 said: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]