​ If the equation (a^2 + b^2) x^2 – 2 (ac + bd) x + c^2 + d^2 = 0 has equal roots, then A. ab = cd B. ad = bc C. ad = √bc ​

Question

If the equation (a² + b²) x² – 2 (ac + bd) x + c² + d² = 0 has equal roots, then

A. ab = cd 

B. ad = bc

C. \(ad=\sqrt{bc}\)

D. \(ab=\sqrt{cd}\)

Answer 1

If the roots are equal then d = b² – 4ac = 0 

Here a = (a² + b²), b = 2 (ac + bd), c = (c² + d²) 

D = b² – 4ac = 0 

⇒ b² = 4ac 

⇒ {– 2(ac + bd)}² = 4{(a2 + b²) (c² + d²)} 

⇒ 4(a²c² + b²d² + 2acbd) = 4(a²c² + a²d² + b²c² + b²d²) 

⇒ 2acbd = a²d² + b²c² 

⇒ a²d² + b²c² – 2abcd = 0 

⇒ (ad – bc)² = 0 

⇒ ad – bc = 0 

⇒ ad = bc