The urns A , B and C contains 4 red, 6 black; 5 red, 5 black and λ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0 . 4 , then the square of length of the side of largest equilateral triangle, inscribed in the parabola y 2 = λ x with one vertex at vertex of parabola is

Step-by-step solution

Let R be the event of drawing red balls. P R A = 4 10 P R B = 5 10 P R C = λ 4 + λ Now, P C R = 0 . 4 ⇒ P C P R C P A P R A + P B P R B + P C P R C = 4 10 ⇒ 1 3 × λ λ + 4 1 3 × 4 10 + 1 3 × 5 10 + 1 3 × λ λ + 4 = 4 10 ⇒ λ λ + 4 9 10 + λ λ + 4 = 4 10 ⇒ λ λ + 4 = 4 10 9 10 + λ λ + 4 ⇒ 5 λ λ + 4 = 2 9 10 + λ λ + 4 ⇒ 5 λ λ + 4 = 19 λ + 36 5 λ + 4 ⇒ 2