Question

If $a,b,c$ are real numbers, then $a(a-b)+b(b-c)+c(c-a)=0,$ only if

• A. $a+b+c=0$
• B. $a=b=c$
• C. Either $a=b$ or $b=c$ or $c=a$
• D. $a-b-c=0$

Answer 1

The correct option is B $a=b=c$

$a(a-b)+b(b-c)+c(c-a)=0$
$\Rightarrow a^2+b^2+c^2-ab-bc-ca=0$
$\Rightarrow 2a^2+2b^2+2c^2-2ab-2bc-2ca=0$
$\Rightarrow (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0$
$\Rightarrow a-b=0$ and $b-c=0$ and $c-a=0$
$\Rightarrow a=b=c$