Question

If $a,b,c$ are non-zero real numbers, then $\left|\begin{array}{ccc}bc& ca& ab\\ ca& ab& bc\\ ab& bc& ca\end{array}\right|$ vanishes, when

• A. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$
• B. $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}=0$
• C. $\frac{1}{b}+\frac{1}{c}-\frac{1}{a}=0$
• D. $\frac{1}{b}-\frac{1}{c}-\frac{1}{a}=0$

Answer 1

Answer: (A)

We have $\left|\begin{array}{ccc}bc& ca& ab\\ ca& ab& bc\\ ab& bc& ca\end{array}\right|=0$
$\Rightarrow 3a^2{b}^2{c}^2-\left[(ab{)}^3+(bc{)}^3+(ca{)}^3\right]=0$
$\Rightarrow (ab{)}^3+(bc{)}^3+(ca{)}^3-3a^2{b}^2{c}^2=0$
$\Rightarrow (ab+bc+ca)\left(a^2{b}^2+b^2{c}^2+c^2{a}^2-a{b}^{2}c-b{c}^{2}a-c{a}^{2}b\right)=0$
$\Rightarrow ab+bc+ca=0$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$