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Question
If a2 + b2 + c2 − ab − bc − ca =0, then
Options
a + b + c
b + c = a
c + a = b
a = b = c
Solution
Given `a^2 +b^2 +c^2 - ab - bc - ca =0`
Multiplying both sides by 2 we get,
`2a^2 +2b^2 +2c^2 -ab - 2bc -2ca = 2 xx 0`
`(a^2 - 2ab +b^2 )+ (b^2 -2bc +c^2) + (c^2 -2ac +a^2)= 0`
`(a-b)^2 +(b-c)^2 + (c-a)^2 =0`
Therefore the sum of positive quantities is zero if and only if each quantity is zero.
`(a-b)=0 ,(b-c) = 0,(c-a) =0`
`a=b,b=c.c =a`
If `a^2 +b^2 +c^2 - ab -bc -ca =0`, then `a=b=c`.
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