Question

Consider the following linear equations $ax+by+cz=0$
$bx+cy+az=0$
$cx+ay+bz=0$

Match the conditions / expressions in Column I with statements in Column II and indicate your answers by darkening the appropriate bubbles in 4 4 matrix given in the ORS.

Answer 1

Let $△=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$=-\frac{1}{2}(a+b+c)[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2]$

(A) If $a+b+c\ne 0$ and $a^2+b^2+c^2=ab+bc+ca$

$\Rightarrow △=0$ and $a=b=c\ne 0$

$\Rightarrow$ the equations represent identical planes.

(B) $a+b+c=0$ and $a^2+b^2+c^2\ne ab+bc+ca$

$\Rightarrow △=0$

$\Rightarrow$the equations have infinitely many solutions.

$ax+by=(a+b)z$

$bx+cy=(b+c)z$

$\Rightarrow (b^2-ac)y=(b^2-ac)z\Rightarrow y=z$

$\Rightarrow ax+by+cy=0\Rightarrow ax=ay\Rightarrow x=y=z.$

(C) $a+b+c\ne 0$ and $a^2+b^2+c^2\ne ab+bc+ca$

$\Rightarrow △\ne 0$

$\Rightarrow$ the equation represent planes meeting at only one point.

(D) $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$

$\Rightarrow a=b=c$
The equation represents the whole of three dimensional space