Question

(i) If the coordinates of the ends of a focal chord of the parabola y² = 4ax are (x₁, y₁) and (x₂, y₂), then prove that x₁x₂ = a² , y¹y² = –4a² . 

(ii) For a focal chord PQ of the parabola y² = 4ax, if SO = l and SQ = l′ then prove that 1/l+1/ l′ =1/a.

Answer 1

Let P(x₁,y₁) = (at₁² , 2at₁) and Q(x₂, y₂) = (at₂² , 2at₂) be two end points of a focal chord. 

P, S, Q are collinear. 

Slope of PS = Slope of QS

From (1)

ii) Let P(at₁² , 2at₁) and Q(at₂² , 2at₂) be the extremities of a focal chord of the parabola, then t₁t₂ = –1 (from(1))