If the equation (a 2 + b 2 ) x 2 – 2 (ac + bd) x + c 2 + d 2 = 0 has equal roots, then

Question

Philoid · Accepted Answer

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