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Let 'S' be the focus and 'l' be the directrix of parabola

let 'Z' be the projection of 'S' on 'l' & A be the mid point of ZS such that AS=AZ=a

therefore A lies on the parabola

Let AY be parallel to 'l' & choose AY as Y-axis &AS as X-axis

A=(0,0) S(a,0) z(-a,0) and equation of directrix l=x=-a=x+a=0

Let p(x,y)be any point on the parabola

therefore SP/PM = 1 (by def)

=SP=PM

√ (x-a)^2 + y^2 = |x+a| / √ 1^2 + 0^2

Squaring on both sides

(x-a)^2 + y^2 = (x+a)^2

x^2-2ax + a^2 + y^2 = x^2 +a^2+2ax

therefore y^2= 4ax which is standard form of the parabola

Hope it helps!!!!!