Question

if the equation (1+m^2 ) x^+ 2m (x+c^2_a^2 =0 has equal roots then show that c^2=a^2(1+m^2)

Answer 1

(1 + m2)x2 + 2 mcx + c2 - a2 = 0 has equal roots

⇒ b2 - 4ac = 0

⇒ (2 mc)2 - 4(1 + m2)(c2 - a2) = 0

⇒ 4m2c2 - 4(c2 - a2 + m2c2 - m2a2) = 0

⇒ 4m2c2 - 4c2 + 4a2 - 4m2c2 + 4m2a2 = 0

⇒  4m2a2 - 4c2 + 4a2 = 0

⇒ m2a2 - c2 + a2 = 0

⇒ a2(1 + m2) - c2 = 0 

⇒ c2  = a^2(1 + m2)

Hence proved.

Answer 2

see this pic for your answer