Q. 144.6( 20 Votes )

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

Answer :

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


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