Mathematics - Determinant Question with Solution | TestHub

Let $\Delta =\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=-\frac{1}{2}(a+b+c)\left[(a-b{)}^2+(b-c{)}^2+(c-a{)}^2\right]$ (A) If $a+b+c\ne 0$ and $a^2+b^2+c^2=ab+bc+ca$ $$\Rightarrow \Delta =0\text{ and }a=b=c\ne 0$$ The equations represents identical planes. (B) $a+b+c=0$ and $a^2+b^2+c^2\ne ab+bc+ca$ $$\Rightarrow \Delta =0$$ The equations have infinitely many solutions. $$\begin{array}{rlrl}& & ax+by& =(a+b)z\\ \Rightarrow & bx+cy& =(b+c)z& \\ \Rightarrow & & \left(b^2-ac\right)y& =\left(b^2-ac\right)z\Rightarrow y=z\\ \Rightarrow & ax+by+cy& =0& \\ & ax& =ay\Rightarrow x=y=z& \end{array}$$ (C) $a+b+c\ne 0$ and $a^2+b^2+c^2\ne ab+bc+ca$ $$\Rightarrow \Delta \ne 0$$ The equation represent planes meeting at only one point. (D) $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$ $$\Rightarrow a=b=c=0$$ The equations represent whole of the three dimensional space.