If α > β > 0 are the roots of the equation a x 2 + b x + 1 = 0 , and lim x → 1 α 1 - cos x 2 + b x + a 2 ( 1 - α x ) 2 1 2 = 1 k 1 β - 1 α , then k is equal to

Step-by-step solution

Since, α and β are the roots of the equation a x 2 + b x + 1 = 0 , therefore 1 α and 1 β would be the roots of x 2 + b x + a = 0 . Let L = lim x → 1 α 1 - cos x 2 + b x + a 2 1 - α x 2 1 2 → 0 0 ⇒ L = lim x → 1 α 2 sin 2 x 2 + b x + a 2 2 1 - α x 2 1 2 ⇒ L = lim x → 1 α sin 2 1 2 x - 1 α x - 1 β α 2 x - 1 α 2 1 2 ⇒ L = lim x → 1 α x - 1 β 2 4 α 2 × sin 1 2 x - 1 α x - 1 β 2