Question

37)If the quadratic equation (1+a2)b2x2+2abcx+(c2-m2)=0 in x has equal roots prove that
I c²=m²(1+a²)​

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 = a2(1 + m2)

Hence proved.

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Answer 2

Answer:

Step-by-step explanation:

Solution :-

Since, the given equation has equal roots, D = 0

So, D ⇒ b² - 4ac = 0

(2abc)² - 4(1 + a²) b² (c² - m²) = 0

⇒ 4a²b²c² - (4b² + 4a²b²) (c² - m²) = 0

⇒ 4a²b²c² - [4b²c² - 4b²m² + 4a²b²c² - 4a²b²m²] = 0

⇒ 4a²b²c² - 4b²c² + 4b²m² - 4a²b²c² + 4a²b²m² = 0

⇒ 4b² [a²m² + m² - c²] = 0

⇒ c² = a²m² + m²

⇒ c² = m²(1 + a²)

Hence Proved.