Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}, z_{2}\) are two complex numbers such that \(\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1\) and \(t z_{1}=k z_{2}\) where \(k \in \mathbb{R}\), then the angle between \(\left(z_{1}-z_{2}\right)\) and \(\left(z_{1}+z_{2}\right)\) is(A) \(\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)\) (B) \(\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)\) (C) \(-2 \tan ^{-1}(k)\) (D) \(2 \tan ^{-1}(k)\)

Step-by-Step Solution

Answer

(D) \(2 \tan^{-1}(k)\)

Step 1: Understand the condition

We are given that \(\left|\frac{z_1 - z_2}{z_1 + z_2}\right| = 1\). This implies that the magnitudes of \((z_1 - z_2)\) and \((z_1 + z_2)\) are equal.

Step 2: Determine magnitudes

Since the magnitudes of \(z_1 - z_2\) and \(z_1 + z_2\) are equal, i.e., \(|z_1 - z_2| = |z_1 + z_2|\), it means the angle between them is \(\frac{\pi}{2}\) due to equal magnitudes for a ratio of 1.

Step 3: Consider the angle formula

The formula for the angle \(\theta\) between two complex numbers \(a\) and \(b\) is \(\tan \theta = \frac{\text{Im}(\overline{a}b)}{\text{Re}(\overline{a}b)}\). Since \(|z_1 - z_2| = |z_1 + z_2|\), solving for \(z_1 = k z_2\) and integrating gives the angle between the two is \(\frac{\pi}{2}\).

Step 4: Verify using condition

From the equation \(tz_1 = k z_2\), we assume \(z_1 = rx\) and \(z_2 = ry\) for a real number \(r\). This still obeys the condition \( |z_1 - z_2| = |z_1 + z_2| \) since arguments \(x\) and \(y\) are symmetric leading to \(2 \tan^{-1}(k)\).

Step 5: Answer verification

Among the options provided, option (D) \(2 \tan^{-1}(k)\) matches that the angle between \((z_1 - z_2)\) and \((z_1 + z_2)\) is determined by the relationship derived from the given conditions.

Key Concepts

Angle between complex numbers

When dealing with complex numbers, understanding the angle between them is crucial for operations like addition or multiplication. To find the angle between two complex numbers, consider them as vectors in the complex plane. Each has a direction determined by its imaginary and real components.

In many cases, especially when given conditions like \[\left| \frac{z_1 - z_2}{z_1 + z_2} \right| = 1,\]we can determine something significant about their directions. Here, this condition implies that the magnitudes of \( z_1 - z_2 \) and \( z_1 + z_2 \) are equal, indicating these vectors are perpendicular. Hence, the angle between them is \( \frac{\pi}{2} \) or 90 degrees.

For any two complex numbers, the angle \( \theta \) between them can be found using the formula:\[\tan \theta = \frac{\text{Im}(\overline{a}b)}{\text{Re}(\overline{a}b)},\]where \( \overline{a} \) represents the complex conjugate of \( a \) and \( b \) is the other complex number. This formula uses both imaginary and real parts to determine their relative orientation.

Magnitude of complex numbers

The magnitude of a complex number is an essential property that represents its size or distance from the origin in the complex plane. Knowing how to calculate the magnitude is a fundamental skill in understanding and manipulating complex numbers.

For a complex number \[z = a + bi,\]its magnitude, denoted as \( |z| \), is calculated using the formula:\[|z| = \sqrt{a^2 + b^2}.\]This results from treating the complex number as a point \((a, b)\) in a plane and finding its distance from the origin, similar to the Pythagorean theorem.

In complex number operations, especially when conditions like \( \left| \frac{z_1 - z_2}{z_1 + z_2} \right| = 1 \) are involved, calculating magnitude is pivotal. Two complex numbers having the same magnitude, as in the exercise, often leads to symmetrical or perpendicular relations, simplifying the analysis of their angles and interactions.

Complex number operations

Complex number operations often extend basic arithmetic operations to accommodate the imaginary unit \( i \). Understanding these operations is foundational to handling complex numbers effectively.

Key operations include:

• Addition: Combine real parts and imaginary parts separately. If \( z_1 = a + bi \) and \( z_2 = c + di \), then \( z_1 + z_2 = (a+c) + (b+d)i \).
• Subtraction: Similar to addition, but subtract the components separately: \( z_1 - z_2 = (a-c) + (b-d)i \).
• Multiplication: Use distributive property, remembering that \( i^2 = -1 \). Thus, \( z_1 \cdot z_2 = (ac-bd) + (ad+bc)i \).
• Division: Involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part of the denominator: \[\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}.\]

These fundamental operations enable us to manipulate complex numbers in various algebraic forms, revealing their geometric relationships and simplifying problem-solving in a complex plane.