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Construct Parabola Given Two Points and Axis of Symmetry

Given two points, one at the origin, one at (a, b) in the first quadrant. Given axis of symmetry, the vertical line at x = x_0 > a. Find the directrix and the vertex (or focus). The given point (a, b) is controlled by the two purple points (a, 0) and (0, b) on the axes, and the axis of symmetry is controlled by the orange point at (x_0, 0). All other points are fixed in relation to the above three objects and cannot be manipulated directly.