Problem 214 - A Complete Resource Book in Mathematics for JEE Main

Question

A complex number \(z\) is said to be unimodular if \(|z|=1 .\) Suppose \(z_{1}\) and \(z_{2}\) are complex numbers such that \(\frac{z_{1}-2 z_{2}}{2-z_{1} \bar{z}_{2}}\) is unimodular and \(z_{2}\) is not unimodular. Then the point \(z_{1}\) lies on a \(|2015|\) (A) straight line parallel to \(y\)-axis. (B) circle of radius \(2 .\) (C) circle of radius \(\sqrt{2}\). (D) straight line parallel to \(x\)-axis.

Step-by-Step Solution

Answer

The point \(z_1\) lies on a circle of radius \(2\).

Step 1: Understand the problem

We need to determine the nature of the geometric place on which the complex number \(z_1\) lies. We are given that the expression \(\frac{z_{1}-2 z_{2}}{2-z_{1} \bar{z}_{2}}\) is unimodular \((|z|=1)\) and that \(z_2\) itself is not unimodular.

Step 2: Modulus condition for unimodularity

Since the complex number \(z\) is unimodular, we have the condition \(|z| = 1\). Thus, for the given expression, the modulus condition becomes: \[ \left| \frac{z_{1}-2 z_{2}}{2-z_{1} \bar{z}_{2}} \right| = 1. \]

Step 3: Simplify the modulus equation

Applying the modulus condition, we express it as: \[ |z_1 - 2z_2| = |2 - z_1 \bar{z}_2|. \]This implies that both parts have the same length (i.e., absolute value in the complex plane).

Step 4: Express in complex parts

Substitute ##z_1 = x + yi## and ##z_2 = a + bi##, where both are complex numbers. Use this to simplify the expression \[ |(x + yi) - 2(a + bi)| = |2 - (x+yi)(a-bi)|. \]

Step 5: Solve for the equations of a circle or line

By squaring the modulus equality, simplifying, and expanding both sides, we are analyzing whether this forms a standard equation for a circle or a line geometrically.

Step 6: Establish result as circle

Upon equating and simplifying using algebra, we find that the expression forms a circle equation, specifically related to the standard form of a circle centered at origin but affecting the radius value.

Step 7: Derive circle's radius

Detailed calculations will show that the arrangement stipulating both conjugate and non-conjugate forms allows us to reveal the circle's determined radius, revealing the solution under \(|z_1-2z_2|=|2-z_1\bar{z}_2|\).

Key Concepts

Unimodular Complex Numbers

Complex numbers are fascinating mathematical constructs that extend our understanding of number systems. A unimodular complex number is simply a complex number with a modulus (or absolute value) equal to 1. This means that if a complex number is represented as \( z = a + bi \), where \( a \) and \( b \) are real numbers, the unimodular condition is satisfied when \( \sqrt{a^2 + b^2} = 1 \).

• Unimodular complex numbers lie on the unit circle in the complex plane.
• This circle is centered at the origin with a radius of 1.
• The condition \(|z| = 1\) ensures the complex number does not stray from this circle.

Understanding unimodular numbers is essential, especially in problems involving the division or transformation of complex numbers. Their unique property of maintaining a modulus of 1 simplifies calculations and aids in visualizing complex numbers geometrically.

Geometric Representation of Complex Numbers

Representing complex numbers geometrically helps build intuition around their properties and operations. A complex number \(z = a + bi\) can be visualized as a point \((a, b)\) on the complex plane. This plane has the real axis (horizontal) and the imaginary axis (vertical), much like a Cartesian coordinate system.

• Each complex number corresponds to a unique point in this plane.
• The distance from the origin to this point is the modulus of the complex number.
• The line segment from the origin to the point makes an angle with the positive real axis, known as the argument.

Using geometry, you can see operations like addition, subtraction, and even multiplication/division as transformations of these points. For example, multiplying by a unimodular complex number rotates a point about the origin without changing its distance from the origin. This geometric visualization provides an excellent foundation for understanding complex number transformations.

Modulus and Argument of Complex Numbers

The modulus and argument are crucial in understanding complex numbers' magnitude and direction. The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). This represents the number's distance from the origin on the complex plane, analogous to the length of the vector from (0,0) to (a,b).

• The argument of a complex number is the angle \(\theta\) formed between the positive real axis and the line representing the complex number in the plane.
• Calculated as \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\), the argument gives the direction of the vector.
• The polar form \( z = |z|(\cos \theta + i\sin \theta) \) relates directly to the modulus and argument.

These concepts play a significant role when dealing with problems involving rotations and scaling of complex numbers, like transforming the complex number into various forms. Modulus and argument provide a compact way to handle complex arithmetic, leading to elegant solutions in many mathematical problems.