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Question #41650

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be the line of intersection of the planes $x - 2y + z = 1$ and $2x + y - z = 2$ . Let $Q$ be the image of the point $P(-1, -1, 8)$ in the line $L$ . If $R(\alpha, \beta, \gamma)$ is the image of $Q$ in the plane $x + y + z = 2$ , then the value of $\alpha^2 + \beta^2 + \gamma^2$ is

Options

A.
$14$
B.
$46$
C.
$78$
D.
$114$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Symmetric vs Non-symmetric form|Image of a point in a plane|Coordinate Axis Projections

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate between non-symmetric and symmetric forms of a line, and sequentially apply reflection formulas for both lines and planes.

The primary trap lies in the coordinate axis projection phase: students often miscalculate the direction vector of the line from the cross product of the plane normals, or incorrectly solve the linear system to find a base point on the line.

Furthermore, a common secondary trap is confusing the formula for the foot of the perpendicular with the reflection formula (omitting the factor of 2), which leads directly to the distractor option 14.

The multi-step nature ensures that any sign-convention error during the projection cascades into an incorrect final distance metric.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Symmetric vs Non-symmetric form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free