Problem 202 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(w=\frac{z}{z-\frac{1}{3} i}\) and \(|w|=1\), then \(z\) lies on \([2005]\) (A) an ellipse (B) a circle (C) a straight line (D) a parabola

Step-by-Step Solution

Answer

(C) a straight line

Step 1: Understand the Given Equation

The equation is given as \( w = \frac{z}{z - \frac{1}{3}i} \) and the condition \( |w| = 1 \). This indicates that \( w \) is a complex number with a magnitude of 1, meaning that it lies on the unit circle in the complex plane.

Step 2: Use the Magnitude Condition

Since the magnitude of \( w \) is 1, we use the property \( |w| = 1 \) which implies \( |z| = |z - \frac{1}{3}i| \). This means the distance from \( z \) to the origin is the same as its distance to the point \( \frac{1}{3}i \) on the imaginary axis.

Step 3: Interpret the Geometrical Condition

The condition that \( |z| = |z - \frac{1}{3}i| \) describes a set of points that are equidistant from the origin and the point \( \frac{1}{3}i \). This is the geometrical definition of a perpendicular bisector and, specifically in the complex plane, describes a straight line.

Key Concepts

Geometrical Interpretation

When dealing with complex numbers, geometrical interpretation helps to visualize solutions on the complex plane. In the problem provided, we see the condition \(|w|=1\), which hints that our complex number \(w\) has a magnitude of 1. This magnitude condition implies that \(w\) lies on the unit circle, a fundamental concept in complex number geometry.
Understanding how lines and distances translate to complex equations is crucial.

• The equation \(|z| = |z - \frac{1}{3}i|\) suggests that the point \(z\) is equidistant from the origin and the point on the imaginary axis \(\frac{1}{3}i\).
• This reflects a core geometric principle: when two distances are equal, the point \(z\) traces a perpendicular bisector. In the context of the complex plane, this translates to a straight line due to the linear nature of our equation.

Magnitude Condition

Magnitude is a key concept when working with complex numbers in mathematical exercises. Here, \(|w| = 1\) plays a central role. Let's break it down:
This condition \(|w| = 1\) signifies that the complex number \(w\) lies precisely on the unit circle, where every point in the complex plane is exactly one unit away from the origin.
In practice, this means:

• The formula \(|z| = |z - \frac{1}{3}i|\) arises. It specifies that the magnitude of \(z\) is equal to the magnitude of \(z - \frac{1}{3}i\).
• In geometric terms, this implies that \(z\) maintains equal distances between the origin and the artificial point \(\frac{1}{3}i\), thus forming a straight line.

Complex Plane

The complex plane, also known as the Argand plane, is essential for visualizing complex numbers. It's a two-dimensional plane where each point corresponds to a complex number, though it harnesses both real and imaginary components.
For this particular problem, consider how \(z\) and \(w\) relate to the complex plane.

• The expression \(w = \frac{z}{z - \frac{1}{3}i}\) uses division, indicating transformation in the plane.
• By focusing on the relationships of these transformations and their magnitudes, the solution ultimately reveals that the set of possible values for \(z\) lies along a straight line in the complex plane, precisely aligning to its geometric characteristics.

This understanding greatly aids in connecting algebraic expressions with their spatial representations in mathematical problems.