Question

If $a+b+c=0$ and $a^2+b^2+c^2=16$, find the value of $ab+bc+ca.$

Answer 1

Given, $a+b+c=0$ and $a^2+b^2+c^2=16,$

Using identity,
$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$
$\Rightarrow (0{)}^2=16+2ab+2bc+2ca$
$\Rightarrow 2ab+2bc+2ca=-16$
$\Rightarrow 2(ab+bc+ca)=-16$
$\therefore ab+bc+ca=-8$

Hence, the value of $ab+bc+ca$ is $-8$.