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Question #40754

Difficulty: 8/10 | 🟢 Free

The Question

A new coordinate system $(X, Y, Z)$ is obtained by shifting the origin of the original system $(x, y, z)$ to a point $S(\alpha, \beta, \gamma)$ without rotating the axes. In this new system, three lines $L_1, L_2, L_3$ are given by the equations: $L_1: \frac{X-2a}{a} = \frac{Y-2b}{b} = \frac{Z-2c}{c}$ $L_2: \frac{X - (a+2b)}{b} = \frac{Y - (b+2c)}{c} = \frac{Z - (c+2a)}{a}$ $L_3: \frac{X - a + c}{c} = \frac{Y - b + a}{a} = \frac{Z - c + b}{b}$ where $a, b, c$ are distinct non-zero real numbers. It is given that the lines $L_1, L_2, L_3$ are coplanar. Let $\Pi$ be the plane containing these three lines. If $\Pi$ intersects the original coordinate axes at points $A, B,$ and $C$ , and the volume of the tetrahedron $OABC$ (where $O$ is the original origin) is $36$ cubic units, then the value of $|\alpha + \beta + \gamma|$ is:

Options

A.
4
B.
6
C.
8
D.
12

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Concurrent Lines Condition|Collinearity vs Coplanarity|Shifting Origin

Why This Question Is Tricky

Common Mistakes

This question tests the invariance of concurrency and coplanarity under origin shifts, specifically targeting the 'Collinearity vs Coplanarity' trap in 3D geometry.

Students often mistakenly apply 2D determinant conditions directly to 3D lines without verifying intersection.

Here, the lines are concurrent at a hidden point $(a,b,c)$ in the shifted frame.

The coplanarity condition forces the direction vector determinant to zero, invoking the identity $a^3+b^3+c^3-3abc=0$ .

Recognizing that $a+b+c=0$ simplifies the seemingly complex normal vector to $(1,1,1)$ .

Failing to account for the origin shift $(\alpha, \beta, \gamma)$ leads to the incorrect plane $x+y+z=0$ , yielding a zero volume and trapping the student.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Concurrent Lines Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free