If two non-horizontal lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate w.r.t. S = 0, then l1n2 + l2n1 = 2am1m2.
Proof : The given non-horizontal lines are
l1x + m1y + n1 = 0 .... (1)
l2x + m2y + n2 = 0 .... (2)
The pole of (1) w.r.t. to the parabola S ≡ y2 – 4ax = 0 is P and the lines (1) and (2) are conjugate w.r.t. S = 0.
i.e., l2 + m2 + n2 = 0
⇒ l1n2 + l2n1 = 2am1m2