Question

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

Answer 1

Given quadratic equation: (1 + m²)x² + 2mcx + (c² – a²) = 0
The given equation has equal roots, therefore

D = b² − 4ac = 0 ..... (1).

From the above equation, we have

a = (1 + m²)

b = 2mc

and c = (c² – a²)

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

D = (2mc)² − 4(1 + m²) (c² – a²) = 0

⇒ 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.