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