Before You Begin

Question #40895

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(1, 1, 1)$ be a point and $\Pi$ be the plane $2x + 2y - z + 6 = 0$ . Let $P'$ be the image of $P$ in the plane $\Pi$ . If the point $Q(\alpha, \beta, \gamma)$ divides the line segment $PP'$ externally in the ratio $3:1$ , then the value of $\alpha + \beta + \gamma$ is:

Options

A.
$-6$
B.
$-78$
C.
$-24$
D.
$6$

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

Section Formula Internally/Externally|Image of a point in a plane|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests the precise application of the 3D point reflection formula and the external section formula.

The primary trap lies in the direction cosines constraint: students frequently forget to normalize the normal vector $\vec{n}$ by dividing by $|\vec{n}|^2$ (or mistakenly divide by $|\vec{n}|$ ) when calculating the image of a point.

If the denominator $a^2+b^2+c^2$ is omitted, the reflection ratio scales incorrectly (yielding $-78$ ), which then cascades into the external section formula, guaranteeing a completely flawed coordinate for $Q$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Section Formula Internally/Externally
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free