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Question #40466

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in $\mathbb{R}^3$ consisting of all points $(x, y, z)$ that lie on the plane $x + y + z = 7$ and satisfy the condition $|(x+iy) - (1+i)| \ge 2\sqrt{2}$ , where $i = \sqrt{-1}$ . If $d$ is the minimum distance from the point $A(1, 1, 0)$ to the region $S$ , then the value of $d$ is:

Options

A.
3
B.
$\frac{5}{\sqrt{3}}$
C.
$\sqrt{17}$
D.
4

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Geometrical Interpretation of Modulus|Distance from point to plane|Locus of a Moving Point|Unbounded Region Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to intersect a 3D plane with a cylindrical constraint derived from a complex modulus inequality.

The primary trap is the 'Unbounded Region Trap' combined with false global optimization.

Students will instinctively calculate the perpendicular distance from the point to the plane ( $5/\sqrt{3}$ ) and assume it is the minimum.

However, the modulus inequality creates a non-convex 'hole' (the interior of a cylinder) in the plane.

Because the perpendicular foot falls inside this excluded region, the true minimum distance must be found on the boundary of the locus using parametric optimization, testing their geometric intuition over blind formula application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Geometrical Interpretation of Modulus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free