Question

If the quadratic equation (1 + m²)x² + 2mcx + (c² – a²) = 0 has equal roots, prove that c² = a²(1 + m²).  

Answer 1

Given: 

(1 + m²)x² + 2mcx + (c² – a²) = 0

Here, 

a = (1 + m²), b = 2mc and c = (c² – a²)

It is given that the roots of the equation are equal; therefore, we have: 

D = 0 

⇒ (b² – 4ac) = 0 

⇒ (2mc)² – 4 × (1 + m²) × (c² – a²) = 0 

⇒ 4m²c² – 4(c² – a² + m²c² – m²a²) = 0 

⇒ 4m²c² – 4c² + 4a² – 4m²c² + 4m²a² = 0 

⇒ –4c² + 4a² + 4m²a² = 0 

⇒ a² + m²a² = c² 

⇒ a²(1 + m²) = c² 

⇒ c² = a²(1 + m²) 

Hence proved.