Let f(x) = (x-p)^2 - q , where p, q R and q 0 . The equation |f(x)| = K (where K 0 ) has exactly four distinct real roots, and these four roots form an arithmetic progression. If the absolute difference between the roots of f(x) = 0 is

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Let f(x) = (x-p)^2 - q , where p, q R and q > 0 . The equation |f(x)| = K (where K > 0 ) has exactly four distinct real roots, and these four roots form an arithmetic progression. If the absolute difference between the roots of f(x) = 0 is 20 , then the value of K is

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Correct answer: 80

The roots of f(x) = 0 are given by (x-p)^2 - q = 0 x = p q . The absolute difference between these roots is 2 q . We are given that 2 q = 20 q = 10 q = 100 . Now, consider the equation |f(x)| = K , which means |(x-p)^2 - q| = K . This gives (x-p)^2 = q - K or (x-p)^2 = q + K . For this equation to have four distinct real roots, we must have q - K > 0 . The roots are p q-K and p q+K . Let the roots in increasing order be x₁, x₂, x₃, x₄ . Since they are symmetric about p , they are: x₁ = p - q+K x₂ = p - q-K x₃ = p +

Let f(x) = (x-p)^2 - q , where p, q R and q > 0 . The equation |f(x)| = K (where K > 0 ) has exactly four distinct real roots, and these four roots form an arithmetic progression. If the absolute difference between the roots of f(x) = 0 is 20 , then the value of K is

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