If $a^{2} + b^{2} + c^{2} - a b - b c - c a = 0 t h e n :$

a = b = c

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a \neq b \neq c

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a = b \neq c

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None of these

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Solution

The correct option is A $a = b = c$
$g i v e n$
$a^{2} + b^{2} + c^{2} - a b - b c - c a = 0$
$\Rightarrow 2 (a^{2} + b^{2} + c^{2} - a b - b c - c a) = 0$
$\Rightarrow 2 a^{2} + 2 b^{2} + 2 c^{2} - 2 a b - 2 b c - 2 c a = 0$
$\Rightarrow (a^{2} - 2 a b + b^{2}) + (b^{2} - 2 b c + c^{2}) + (c^{2} - 2 a c + a^{2}) = 0$
$\Rightarrow {(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2} = 0$
$sin c e the sum of square is zero$
$Hence$
$a-b=0,b-c=0,c-a=0$
$a=b=c$

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