SATHEE: 3D Geometry 3 Question 62

3D Geometry 3 Question 62

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

(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. $A \to r, B \to q; C \to p; D \to s$

Solution:

Let $Δ = | \begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array} |$

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

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

$\begin{aligned} (\Rightarrow & Δ & = 0 \\ ) a n d & (a & = b = c \neq 0 \end{aligned}$ )

$\Rightarrow$ The equations represent identical planes.

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

$\Rightarrow Δ = 0$

$\Rightarrow$ The equations have infinitely many solutions.

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

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

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

$\Rightarrow Δ \neq 0$

The equations represent planes meeting at only one point.

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

$\Rightarrow a = b = c = 0$

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

3D Geometry 3 Question 62

Analytical & Descriptive Questions

Answer:

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3D Geometry 3 Question 62

Analytical & Descriptive Questions

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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