If the roots of the quadratic equation in x (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal, prove that ad = bc.

Given that Roots of equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal

Concept Used

If b² – 4ac = 0 then roots of the quadratic equation ax² + bx + c = 0 are equal

Since the roots are equal,

b² – 4ac = 0 …(i)

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

Putting the values in eq.(i) , we get

[–2(ac + bd)]² – 4(a² + b²)×(c² + d²) = 0

⇒ 4(ac + bd)² – 4a²c² – 4a²d² – 4b²c² – 4b²d² = 0

⇒ 4(a²c² + b²d² + 2abcd) – 4a²c² – 4a²d² – 4b²c² – 4b²d² = 0

⇒ 4a²c² + 4b²d² + 8abcd – 4a²c² – 4a²d² – 4b²c² – 4b²d² = 0

⇒ – 4a²d² – 4b²c² + 8abcd = 0

⇒ – 4(a²d² + b²c² – 2abcd) = 0

⇒ a²d² + b²c² – 2abcd = 0

⇒ (ad – bc)² = 0

⇒ ad – bc = 0

⇒ ad = bc

Hence Proved