Problem 115 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(z_{1}\) and \(z_{2}\) be complex numbers such that \(z_{1} \neq z_{2}\) and \(\left|z_{1}\right|=\left|z_{2}\right| .\) If \(z_{1}\) has positive real part and \(z_{2}\) has negative imaginary part, then \(\frac{z_{1}+z_{2}}{z_{1}-z_{2}}\) may be(A) 0 (B) real and positive (C) real and negative (D) purely imaginary

Step-by-Step Solution

Answer

(D) purely imaginary.

Step 1: Understanding the Problem

We are given two complex numbers, \(z_1\) and \(z_2\), with equal magnitudes, \(|z_1| = |z_2|\), but different arguments such that the real part of \(z_1\) is positive and the imaginary part of \(z_2\) is negative. We need to find the nature of the number \(\frac{z_1 + z_2}{z_1 - z_2}\).

Step 2: Express Complex Numbers in Polar Form

Express \(z_1 = r e^{i\theta} = x + yi\) and \(z_2 = r e^{i\phi} = u + vi\) where \(r\) is the common magnitude. Given the conditions, \(x > 0\) and \(v < 0\).

Step 3: Simplify the Expression

To simplify \(\frac{z_1 + z_2}{z_1 - z_2}\), use \(z_1 = x + yi\) and \(z_2 = u + vi\):\[\frac{(x + u) + (y + v)i}{(x - u) + (y - v)i}\]

Step 4: Conjugate and Rationalize the Denominator

Multiply numerator and denominator by the conjugate of the denominator:\[\frac{[(x + u) + (y + v)i]((x - u) - (y - v)i)}{(x-u)^2 + (y-v)^2}\]This will simplify into:\[\frac{(x+u)(x-u) - (y+v)(y-v) + [(x+u)(y-v) + (y+v)(x-u)]i}{(x-u)^2 + (y-v)^2}\]

Step 5: Identify Potential Outcomes

The expression is purely imaginary if the real part of the numerator is zero, i.e., \[(x+u)(x-u) = (y+v)(y-v)\], leading to both the numerator and denominator being real numbers.

Step 6: Choose the Correct Option

Given equal magnitudes and opposing conditions for the real and imaginary parts, \(\frac{z_1+z_2}{z_1-z_2}\) resolves to being purely imaginary when the real components are effectively canceled out due to symmetry, making only the calculation relevant for the imaginary part. Thus, option (D) is correct.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. Complex numbers take the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. In the complex plane, the horizontal axis (often called the real axis) represents the real part of the number, while the vertical axis (imaginary axis) represents the imaginary part.


- **Visual Representation**: A complex number \( z = x + yi \) is represented as a point \( (x, y) \) on the plane, or as a vector from the origin to this point.
- **Magnitude**: The magnitude, or modulus, of \( z \) is the length of the vector \( |z| = \sqrt{x^2 + y^2} \). For two complex numbers, if their magnitudes are equal, they lie on a circle centered on the origin with radius equal to that magnitude.

Understanding the complex plane helps in visualizing operations like addition, subtraction, and finding conjugates, as well as solving complex equations geometrically.

Polar Form

Polar form is an alternative representation of complex numbers, emphasizing the magnitude of the complex number and its angle with the real axis. Instead of using Cartesian coordinates, it uses radius and angle.

- **Conversion**: From \( z = x + yi \), you convert to polar form \( z = r (\cos \theta + i \sin \theta) \) or \( z = r e^{i\theta} \), where \( r = \sqrt{x^2 + y^2} \) is the magnitude and \( \theta \) (theta) is the angle from the positive real axis.
- **Uses**: The polar form is especially useful for multiplying complex numbers, as magnitudes multiply and angles add: \( z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} \).

In problems like our initial example, using the polar form can often simplify calculations and help to see the symmetry between the numbers more clearly. This makes identifying solutions more intuitive.

Conjugate

The conjugate of a complex number is a concept used to simplify division among complex numbers. For a given complex number \( z = x + yi \), its conjugate is denoted as \( \bar{z} = x - yi \).

- **Properties**:
- Conjugates have the same real part but opposite imaginary parts.
- Multiplying a complex number by its conjugate results in a real number: \( z \bar{z} = (x + yi)(x - yi) = x^2 + y^2 \).
- **Applications**: When dividing complex numbers, the conjugate is used to rationalize the denominator, converting it into a real number so that calculations simplify.

In our example, rationalizing the denominator by multiplying the numerator and the denominator by the conjugate helps in achieving potential outcomes, specifically identifying purely imaginary results by focusing on canceling out real parts.