Consider the following linear equations a x+b y+c z=0, b x+c y+a...

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 \neq 0 and $a^2+b^2+c^2=a b+b c+c a$ \Rightarrow \quad \Delta=0 $\text{\text{ and } }$ a=b=c \neq 0 The equations represents identical planes. (B) $a+b+c=0$ and $a^2+b^2+c^2 \neq a b+b c+c a$ \Rightarrow \quad \Delta=0 The equations have infinitely many solutions. $\begin{array}{rlrl} & & a x+b y & =(a+b) z \\ \Rightarrow & b x+c y & =(b+c) z \\ \Rightarrow & & \left(b^2-a c\right) y & =\left(b^2-a c\right) z \Rightarrow y=z \\ \Rightarrow & a x+b y+c y & =0 \\ & a x & =a y \Rightarrow x=y=z \end{array}$ (C) $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq a b+b c+c a$ \Rightarrow \quad \Delta \neq 0 The equation represent planes meeting at only one point. (D) $a+b+c=0$ and $a^2+b^2+c^2=a b+b c+c a$ \Rightarrow \quad a=b=c=0 The equations represent whole of the three dimensional space.