Before You Begin

Question #40780

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{z \in \mathbb{C} : z^{2024} = 1\}$ . Let $z_1, z_2 \in S$ be two distinct complex numbers such that $z_1 + z_2 \neq 0$ . Let $z_0 = \frac{2}{\bar{z}_1 + \bar{z}_2}$ . Let $L$ be the line in the complex plane passing through $z_0$ and $z_1$ . If $z_3 \in S$ is a point lying on the line $L$ , then the value of $\sum_{k=0}^{10} \left( \frac{z_3}{z_1} \right)^k$ is:

Options

A.
0
B.
1
C.
10
D.
11

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Condition of Collinearity|Polar and Pole Duality|Roots of Unity Properties

Why This Question Is Tricky

Common Mistakes

This question tests the duality of pole and polar intertwined with the modulus properties of roots of unity.

The trap lies in the collinearity condition defining the line $L$ .

Students often attempt to solve the collinearity equation $\text{Im}((z_3-z_1)/(z_0-z_1)) = 0$ algebraically, getting lost in complex expansions.

The conceptual breakthrough requires recognizing that $z_0$ is the pole of the secant $z_1 z_2$ , making it the intersection of tangents at $z_1$ and $z_2$ .

Consequently, the line $z_0 z_1$ collapses into the tangent at $z_1$ .

The modulus constraint $|z_3|=1$ then forces $z_3$ to be the point of tangency itself ( $z_3=z_1$ ), catching students who ignore the geometric implications of the periodic complex roots.

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