3D Geometry 3 Question 62

####62. Consider the following linear equations $a x+b y+c z=0, \quad b x+c y+a z=0, \quad c x+a y+b z=0$

(IIT 2007, 6M)

Column I Column II
A. $a+b+c \neq 0$ and
$a^{2}+b^{2}+c^{2}$
$=a b+b c+c a$
p. The equations represent
planes meeting only at
a single point
B. $a+b+c=0$ and
$a^{2}+b^{2}+c^{2}$
$\neq a b+b c+c a$
q. The equations represent
the line $x=y=z$
C. $a+b+c \neq 0$ and
$a^{2}+b^{2}+c^{2}$
$\neq a b+b c+c a$
r. The equations represent
identical planes
D. $a+b+c=0$ and
$a^{2}+b^{2}+c^{2}$
$=a b+b c+c a$
s. The equations represent
the whole of the
three-dimensional space

Analytical & Descriptive Questions

Show Answer

Answer:

Correct Answer: 62. $A \rightarrow r, B \rightarrow q ; C \rightarrow p ; D \rightarrow s$

Solution:

  1. 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 $a^{2}+b^{2}+c^{2}=a b+b c+c a$

$ \begin{aligned}(\Rightarrow & \Delta & =0 \\) and & (a & =b=c \neq 0\end{aligned} $)

$\Rightarrow$ The equations represent identical planes.

B. $a+b+c=0$ and $a^{2}+b^{2}+c^{2} \neq a b+b c+c a$

$\Rightarrow \Delta=0$

$\Rightarrow$ The equations have infinitely many solutions.

$ a x+b y=(a+b) z, \quad b x+c y=(b+c) z $

$ \begin{aligned} & \Rightarrow \quad\left(b^{2}-a c\right) y=\left(b^{2}-a c\right) z \Rightarrow y=z \\ & \Rightarrow \quad a x+b y+c y=0 \Rightarrow a x=a y \Rightarrow x=y=z \end{aligned} $

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 equations represent planes meeting at only one point.

D. $a+b+c=0$ and $a^{2}+b^{2}+c^{2}=a b+b c+c a$

$\Rightarrow \quad a=b=c=0$

$\Rightarrow$ The equations represent whole of the three-dimensional space.



NCERT Chapter Video Solution

Dual Pane