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

Given: (1 + m²) x² + 2mcx + c² – a² = 0

To prove: c² = a (1 + m²)

Proof: it is being that equation has equal roots, therefore

D = b² – 4ac = 0 …(1)

From the equation, we have

a = (1 + m²), b = 2mc, c = c² – a²

putting values of a, b and c in (1), we get

D = (2mc)² – 4(1 + m²)(c² – a²)

⇒ 4m²c² – 4(c² + c²m² – a² – a²m²) = 0

⇒ 4m²c² – 4c² – 4c²m² + 4a² + 4a²m² = 0

⇒ –4c² + 4a² + 4a²m² = 0

⇒ 4c² = 4a² + 4a²m²

⇒ c² = a² + a²m²

⇒ c² = a² (1 + m²)

Hence proved.