Question

If the roots of the equation (a² b²) x² – 2b (a + c) x + (b² + c²) = 0 are equal, then

A. 2b = a + c 

B. b² = ac

C. \(b=\frac{2ac}{a\,+\,c}\)

D. b = ac

Answer 1

The roots of the equation are equal so d = b² – 4ac = 0 

Here a = (a² + b²), b = – 2b (a + c), c = (b² + c²) 

d = (– 2b (a + c))² = 4 (a² + b²) (b² + c²) 

⇒ b² (a² + 2ac + c²) = a²b² + a²c² + b⁴ + b²c² 

⇒ (ac)² – 2(ac) (b²) + (b²)² = 0 

⇒ (ac – b²)² = 0 

⇒ (ac – b²) = 0 

⇒ a c = b²