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Question #40884

Difficulty: 8/10 | 🟢 Free

The Question

The equation of a plane $\Pi$ in the original coordinate system $Oxyz$ is $2x - y + 2z + 11 = 0$ . The origin is shifted to the point $(1, 2, -1)$ without rotating the axes, establishing a new coordinate system $O'XYZ$ . A point $P$ has coordinates $(3, 1, 2)$ in the new system $O'XYZ$ . Find the coordinates of the image of $P$ in the plane $\Pi$ with respect to the new coordinate system $O'XYZ$ .

Options

A.
$(-5, 5, -6)$
B.
$(-4, 7, -7)$
C.
$(-1, 3, -2)$
D.
$\left(-\frac{53}{9}, \frac{49}{9}, -\frac{62}{9}\right)$

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

Projection vs. Coordinate|Image of a point in a plane|Shifting Origin

Why This Question Is Tricky

Common Mistakes

This question tests the integration of 3D coordinate geometry with origin transformations.

The primary trap lies in the 'Shifting Origin' concept.

Students often calculate the image using the new point coordinates $(3, 1, 2)$ but mistakenly use the old plane equation $2x - y + 2z + 11 = 0$ , forgetting that the constant 'd' and the variables must be transformed to the new frame $O'XYZ$ (leading to Option D).

Another common error is finding the correct image in the old coordinate system $(-4, 7, -7)$ but failing to translate it back to the new system (Option B).

Option C represents the foot of the perpendicular rather than the image.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free