Let P ( α , β ) be a point on the parabola y 2 = 4 x . If P also lies on the chord of the parabola x 2 = 8 y whose mid point is 1 , 5 4 , then ( α - 28 ) ( β - 8 ) is equal to _______.

Step-by-step solution

Given, Parabola x 2 = 8 y We know that, Chord with mid point x 1 , y 1 is T = S 1 x x 1 - 8 y + y 1 2 = x 1 - 2 8 y 1 ⇒ x x 1 - 4 y + y 1 = x 1 - 2 8 y 1 Now, putting the point x 1 , y 1 = 1 , 5 4 we get, ⇒ x - 4 y + 5 4 = 1 - 8 × 5 4 = - 9 ∴ x - 4 y + 4 = 0 … i ( α , β ) lies on (i) and also on y 2 = 4 x ∴ α - 4 β + 4 = 0 … ii And β 2 = 4 α … iii Solving (ii) and (iii) β 2 = 4 ( 4 β - 4 ) ⇒ β 2 - 16 β + 16 = 0 ⇒ β = 8 ± 4 3 And α = 4 β - 4 = 28 ± 16 3 ∴ α , β = 28 + 16 3 , 8 + 4 3 and ( 28 - 16 3 , 8 - 4 3 ) Now,