Before You Begin

Question #40777

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all complex numbers $z$ such that the points representing $z, 1+i,$ and $-1-i$ are collinear on the Argand plane, and $\left[ |z|^2 - 6|z| + 8 \right] = 0$ , where $[ \cdot ]$ denotes the greatest integer function. The total length of the locus of $z$ is:

Options

A.
$4\sqrt{2} - 4$
B.
$2\sqrt{2} - 2$
C.
$4\sqrt{3} - 4\sqrt{2}$
D.
$4\sqrt{2} - 2$

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

Condition of Collinearity|Locus Problems via Transformation|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of geometric collinearity conditions with algebraic interval analysis via the Greatest Integer Function (GIF).

The primary trap lies in the geometric interpretation of the locus.

Students often correctly solve the GIF inequality to find the valid intervals for the modulus $|z|$ (yielding a length of $2\sqrt{2}-2$ ).

However, they fail to recognize that collinearity with $1+i$ and $-1-i$ defines a full line through the origin, not just a single ray.

Consequently, the valid distances $r=|z|$ apply symmetrically in both the first and third quadrants, requiring the interval length to be doubled.

Option B is the direct result of falling into this trap.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Condition of Collinearity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free