Standard equation of a parabola:
Let S be focus, ZM the directrix and P moving point. Draw SZ perpendicular from S on directrix. Then SZ is the axis of the parabola. Now middle point of SZ, say A, will lie on the locus of P, that is, AS = AZ. Take A as the origin, x-axis along AS, and y- axis along the perpendicular to AS at A, as in figure.
Let AS = a, such that ZA is also a. Let (x, y) be the coordinates of moving point P. Then MP = ZN = ZA + AN = a + x. But by the definition MP = PS => MP2 = PS2
So that, (a + x)2 = (x - a)2 + y2.
Therefore, the equation of parabola is y2 = 4ax.
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