If the quadratic equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has eq

1.	If the quadratic equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has equal roots, prove that c2 = a2 (1 + m2).
Answer» (1 + m²) x² + 2mcx + c² – a² = 0 Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0 a = (1 + m²), b = 2mc and c = c² – a² Since roots are equal, so D = 0 (2mc)² – 4.(1 + m²)(c²– a²) = 0 4 m²c² – 4c² + 4a² – 4 m²c² + 4 m²a² = 0 a² + m²a² = c² or c² = a² (1 + m²) Hence Proved

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

Answer»

(1 + m²) x² + 2mcx + c² – a² = 0

Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0

a = (1 + m²), b = 2mc and c = c² – a²

Since roots are equal, so D = 0

(2mc)² – 4.(1 + m²)(c²– a²) = 0

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

a² + m²a² = c²

or c² = a² (1 + m²)

Hence Proved