If a² + b² + c² - ab - bc - ca

Q: If a² + b² + c² - ab - bc - ca <= 0, where a, b, c are non-zero real numbers, then the value of (a + b + c) / c is:

3. The identity a² + b² + c² - ab - bc - ca = (1/2)[(a-b)² + (b-c)² + (c-a)²] shows it is always non-negative. Combined with the given condition (<= 0), it must be exactly 0, which requires a = b = c. Therefore (a+b+c)/c = 3c/c = 3.

Question

If a² + b² + c² - ab - bc - ca <= 0, where a, b, c are non-zero real numbers, then the value of (a + b + c) / c is:

Accepted Answer

3. The identity a² + b² + c² - ab - bc - ca = (1/2)[(a-b)² + (b-c)² + (c-a)²] shows it is always non-negative. Combined with the given condition (<= 0), it must be exactly 0, which requires a = b = c. Therefore (a+b+c)/c = 3c/c = 3.

JEE Advanced Maths: Algebra — Inequalities and Identities questions with solutions

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