If for some p , q , r ∈ R , all have positive sign, one of the roots of the equation p 2 + q 2 x 2 - 2 q p + r x + q 2 + r 2 = 0 is also a root of the equation x 2 + 2 x - 8 = 0 , then q 2 + r 2 p 2 is equal to-

Step-by-step solution

Given, p 2 + q 2 x 2 - 2 q p + r x + q 2 + r 2 = 0 On simplifying we get, p x - q 2 + q x - r 2 = 0 ⇒ p x - q = 0   &   q x - r = 0 ⇒ x = q p = r q ⇒     x = q p = r q = 4 [because roots of equation x 2 - 2 x - 8 = 0 are 4 or - 2 ] As p , q , r are positive, so x must be 4 . Now, q = 4 p and r = 4 q = 16 p So, q 2 + r 2 p 2 = 4 p 2 + 4 × 4 p 2 p 2 = 16 + 256 = 272 .