Question

If the equation (1 m^2)x^2 2mcx c^2-a^2=0 has equal roots , If the equation (1 m^2)x^2 2mcx c^2-a^2=0 has equal roots , show that c^2=a^2(1 m^2)​

Answer 1

Answer:

Step-by-step explanation:

(1 + m²) x² + 2 m c x + c² - a² = 0

(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  = a2(1 + m2)

Hence proved.

Answer 2

Answer:

1 + m²) x² + 2 m c x + c² - a² = 0

(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  = a2(1 + m2)

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