If polar of P(x1, y1) w.r.t. S = 0 passes through Q(x2, y2), then the polar of Q w.r.t. S = 0 passes through P.
Proof : Let the equation of the parabola be S ≡ y2 – 4ax = 0
∴ Equation of the polar of P(x1, y1) with respect to S = 0 is S1 ≡ yy1 – 2a(x + x1) = 0
If it passes through Q(x2, y2), then S21 ≡ y2y1 – 2a(x2 + x1) = 0
⇒ S12 ≡ y1y2 – 2a(x1 + x2) = 0
i.e., the point P(x1, y1) lies on the line yy2 – 2a(x + x2) = 0 which is a polar of Q(x2, y2) w.r.t. S = 0. Hence the polar of Q w.r.t. S = 0 passes through P.
Note:
P(x1, y1) and Q(x2, y2) are conjugate points w.r.t. S = 0 iff S12 = 0.