For p , q ∈ R , consider the real valued function f x = x - p 2 - q , x ∈ R and q > 0 . Let a 1 , a 2 , a 3 and a 4 be in an arithmetic progression with mean p and positive common difference. If f a i = 500 for all i = 1 , 2 , 3 , 4 , then the absolute difference between the roots of f x = 0 is

Step-by-step solution

Given, f x = 0 ⇒ x - p 2 - q = 0 So, roots are p + q , p - q Now absolute difference between roots will be 2 q . Now given a 1 , a 2 , a 3 , a 4 are in A . P and its mean is p Now let a 1 , a 2 , a 3 , a 4 be a 1 = p - 3 d ,   a 2 = p - d ,   a 3 = p + d   &   a 4 = p + 3 d Now given f a i = 500 So, f a 4 = 500 ⇒ a 4 - p 2 - q = 500 ⇒ a 4 - p 2 - q = 500 ⇒ 9 d 2 - q = 500           . . . . 1 And using f a i = 500   ∀   i = 1 ,