Question

If (a+b+c)² = 3(ab + bc + ca), then which of the following is true?
(1) a #b#c
(2) a > b>c
(3) a <b<c
(4) a = b = c​

Answer 1

answer : option (4) a = b = c

before going to solve the problem, let's remind important formula related to it.

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca ).

so, a³ + b³ + c³ - 3abc = 0 when

1. a + b + c = 0 where a ≠ b = c
2. (a² + b² + c² - ab - bc - ca) = 0 where a = b = c

given, (a + b + c)² = 3(ab + bc + ca)

⇒a² + b² + c² + 2(ab + bc + ca) = 3(ab + bc + ca)

⇒a² + b² + c² - ab - bc -ca = 0

from condition - 2 , it is clear that a = b = c

hence option (4) is correct choice.

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