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
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Answer:
Correct Answer: 62. $2 x-y+z-3=0 ; 62 x+29 y+19 z-105=0$
Solution:
- 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.
