The equation of the polar of the point P(x1, y1) w.r.t. parabola S = 0 is S1 = 0.
Proof : Let the equation of the parabola be S ≡ y2 – 4ax = 0 and P(x1, y1) be a point in the plane of the parabola. Any chord through P meets the parabola in D and E. Let the tangents at D, E to the parabola meet at Q(h, k).
∴ , the chord of contact of Q(h, k) w.r.t. the parabola S = 0 whose equation is yk – 2a(x + h) = 0 passes through P(x1, y1)
∴ y1k – 2a(x1 + h) = 0
Then the point Q(h, k) satisfies the equation
yy1 – 2a (x + x1) = 0
∴ The equation of polar of P(x1, y1) w.r.t. S = 0 is S1 = 0.