Let PQ be a double ordinate of the parabola, y² = 8x, where P lies in the second quadrant. If R divides PQ in the ratio 2:1, then the locus of R is

Let the coordinates of P, Q and R be P(-at₁², 2at₁), Q(-at₁², -2at₁) and R(h, k). As R divides PQ in the ratio 2:1, then
h = (-at₁² + 2(-at₁²))/(2 + 1) = -at₁²
and k = (2(2at₁) + 1(-2at₁))/(2 + 1) = 2at₁/3
From (1) and (2),
t₁² = -h/a and t₁ = 3k/(2a)
Substituting t₁ from (2) into (1),
(3k/(2a))² = -h/a
9k²/4a² = -h/a
9k² = -4ah
Since y² = 8x, then a = 2
9k² = -4(2)h
9k² = -8h
9y² = -8x
The equation of the parabola is y² = 8x. Let P(-at₁², 2at₁) and Q(-at₁², -2at₁). R divides PQ in the ratio 2:1. Then the coordinates of R are given by:
h = (2(-at₁²) + 1(-at₁²))/(2 + 1) = -at₁²
k = (2(2at₁) + 1(-2at₁))/(2 + 1) = (2at₁)/3
From y² = 8x, a = 2. Therefore, h = -2t₁² and k = (4t₁)/3.
t₁² = -h/2 and t₁ = 3k/4
Substituting into t₁² = -h/2:
(3k/4)² = -h/2
9k²/16 = -h/2
9k² = -8h
9y² = -8x
However, since P is in the second quadrant, and R is between P and Q, R must also be in the second quadrant. The correct equation is 9y² = 8x