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Question #40887

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ . A surface $S$ is defined by the locus of $\vec{r}$ satisfying $|\vec{r} - (\hat{i} + \hat{j} + \hat{k})|^2 + |\vec{r} - (3\hat{i} + 3\hat{j} + 5\hat{k})|^2 = 44$ If $S'$ is the image of the surface $S$ reflected across the plane $2x - y + 2z = 17$ , then the equation of $S'$ is:

Options

A.
$|\vec{r} - (6\hat{i} + 7\hat{k})| = 4$
B.
$|\vec{r} - (6\hat{i} + 7\hat{k})| = 16$
C.
$|\vec{r} - (4\hat{i} + \hat{j} + 5\hat{k})| = 4$
D.
$|\vec{r} - (8\hat{i} - 2\hat{j} + 9\hat{k})| = 16$

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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

Distance Formula in 3D|Image of a point in a plane|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the geometrical interpretation of vector/complex modulus equations and the properties of isometric transformations.

The trap lies in attempting to algebraically expand the locus equation into Cartesian coordinates and substitute the reflection transformation, which is extremely time-consuming and prone to error.

The optimal approach requires recognizing the locus as a sphere using Apollonius's theorem (or completing the square), identifying that reflection preserves the radius (modulus constraint), and solely reflecting the geometric invariant—the center of the sphere—across the plane.

Option C traps students who calculate the foot of the perpendicular instead of the image.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Distance Formula in 3D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free