Question

$\text{If}{a}^2+b^2+c^2-ab-bc-ca=0,\text{then}$

• A. a + b = c
• B. b + c = a
• C. c + a = b
• D. a = b = c

Answer 1

The correct option is D

a = b = c

$a^2+b^2+c^2-ab-bc-ca=02(a^2+b^2+c^2-ab-bc-ca)=0\left(\text{Multiplying by 2}\right)\Rightarrow 2a^2+2b^2+2c^2-2ab-2bc-2ca=0\Rightarrow a^2+b^2-2ab+b^2+c^2-2bc+c^2+a^2-2ca=0\Rightarrow (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0\therefore (a-b{)}^2=0,\text{then}a-b=0\Rightarrow a=b\text{Similarly,}(b-c{)}^2=0,\text{then}b-c=0\Rightarrow b=c\text{and}(c-a{)}^2=0,\text{then}c-a=0\Rightarrow c=a\therefore a=b=c$