Question

If three unequal positive real numbers satisfying b^2=ac and b – c, c – a, a – b are in H.P. then the value of a + b + c is independent of​

Answer 1

Given : If three unequal positive real numbers satisfying b² = ac and b – c, c – a, a – b are in H.P .

To find : The value of (a + b + c) is independent of

solution : we have , b – c, c – a, a – b are in H.P .

so, 1/(b - c) , 1/(c - a) and 1/(a - b) are in A.P

⇒1/(c - a) - 1/(b - c) = 1/(a - b) - 1/(c - a)

⇒2/(c - a) = 1/(b - c) + 1/(a - b)

⇒2/(c - a) = [(a - b) + (b - c)]/(b - c)(a - b)

⇒2/(c - a) = (a - c)/(b - c)(a - b)

⇒2(b - c)(a - b) = -(a - c)²

⇒2[ab - b² - ac + bc ] = - [a² + c² - 2ac ]

⇒2ab - 2b² - 2ac + 2bc = - a² - c² + 2ac

⇒a² + c² + 2ab - 2b² - 4ac + 2bc = 0

⇒a² + b² + c² + 2ab + 2bc + 2ca - 6ca - 3b² = 0

⇒(a + b + c)² - 6ca - 3b² = 0

⇒(a + b + c)² - 6b² - 3b² = 0 [ as given b² = ca]

⇒(a + b + c)² = 9b²

⇒(a + b + c) = 3b

Hence the value of (a + b + c) is independent of a and c.

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