If the quadratic equation (1 + m2)x2 + 2mcx + c2 – a2 = 0 has equal roots, prove that c2 = a2(1 + m2) .
The quadratic equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has equal roots
Comparing with standard quadratic equation ax2 + bx + c = 0
a = (1 + m2) b = 2mc c = c2 – a2
Thus D = 0
Discriminant D = b2 – 4ac = 0
(2mc)2 – 4.(1 + m2)(c2 – a2) = 0
4 m2c2 – 4c2 + 4a2 – 4 m2c2 + 4 m2a2 = 0
– 4c2 + 4a2 + 4m2a2 = 0
a2 + m2a2 = c2
c2 = a2 (1 + m2)
Hence proved