Problem 161 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}=a+i b\) and \(z_{2}=c+i d\) are two complex numbers such that \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(\operatorname{Re}\left(z_{1}, \bar{z}_{2}\right)=0\) then for the pair of complex numbers \(\omega_{1}=a+i c\) and \(\omega_{2}=b+i d\) (A) \(\operatorname{Re}\left(\omega_{1} \bar{\omega}_{2}\right)=0\) (B) \(\operatorname{Re}\left(\omega_{1} \bar{\omega}_{2}\right)=1\) (C) \(\left|\omega_{1}\right|=1\) (D) None of these

Step-by-Step Solution

Answer

Due to complex arrangement and constraints, none of the options are strictly verified; answer is (D).

Step 1: Understand the given conditions

We are given two complex numbers, \(z_1 = a + ib\) and \(z_2 = c + id\), with the conditions \(|z_1| = |z_2| = 1\) and \(\text{Re}(z_1 \bar{z}_2) = 0\). This implies that \(z_1\) and \(z_2\) are unit complex numbers on the complex plane, or in other words, they lie on the unit circle.

Step 2: Interpreting Re(z1 z2) = 0

The expression \(\text{Re}(z_1 \bar{z}_2) = 0\) means that the real part of the product of \(z_1\) and the complex conjugate of \(z_2\) is zero, indicating that this product is purely imaginary.

Step 3: Determine properties of ω1 and ω2

Define \(\omega_1 = a + ic\) and \(\omega_2 = b + id\). We need to analyze the expression \(\text{Re}(\omega_1 \bar{\omega}_2)\) to see if it satisfies any of the given options.

Step 4: Compute ω1 and ω2 product

Calculate \(\omega_1 \bar{\omega}_2 = (a + ic)(b - id) = ab + ad + i(-bd + bc)\). Then, the real part is \(ab + ad\).

Step 5: Set Re(ω1 ω̅2) based on given conditions

Given the condition \(\text{Re}(z_1 \bar{z}_2)=0\), we have \(a \cdot c + (-b) \cdot d = 0\), which simplifies to \(ac - bd = 0\). This helps in determining the result of \(\omega_1 \bar{\omega}_2\). Since \(ab + ad\) needs explicit computation to verify zero, only possible when \(ad + bc = 0\) and \(ab = 1\).

Step 6: Conclusion for ω1 ω̅2 real part

Since \(ac - bd = 0\), \(\omega_1 \bar{\omega}_2 = ab + bc\) is not strictly 0 or 1; the real part doesn't satisfy given options based on this analysis prior conditions.

Key Concepts

Unit Circle

When dealing with complex numbers, the unit circle plays a significant role. It is a circle with a radius of 1 centered at the origin of the complex plane. The complex plane itself is a two-dimensional plane where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.

For a complex number to be on the unit circle, its magnitude or modulus must be equal to 1. This condition is given by the equation: \[ |z| = 1\] where \( z = a + ib \).

• If \( |z_1| = 1 \)and \( |z_2| = 1 \), both complex numbers \( z_1 \) and \( z_2 \)lie on the unit circle.

Understanding the unit circle is crucial in this exercise, as it tells us that both complex numbers \( z_1 \)and \( z_2 \)have points equidistant from the origin, making them pure rotations around the circle.

Complex Conjugate

The complex conjugate of a complex number is a reflection of the number across the real axis. For any complex number \( z = a + ib \), its complex conjugate is denoted as \( \bar{z} = a - ib \).

This operation changes the sign of the imaginary part while leaving the real part unchanged. In this problem, \( z_1 \)and \( ar{z}_2 \)are used together to form a special condition.

• Given \( ext{Re}(z_1 ar{z}_2) = 0 \), the condition indicates that the product \( z_1 ar{z}_2 \) is purely imaginary.

This requirement implies the vectors corresponding to \( z_1 \)and \( z_2 \)take a perpendicular orientation on the complex plane. Understanding complex conjugates is essential for realizing how complex numbers interact with each other under multiplication.

Imaginary Part

The imaginary part of a complex number is the coefficient of the imaginary unit \( i \). In a complex number \( z = a + ib \), \( b \)is considered the imaginary part.

In this context, when multiplying complex numbers like \( ar{z}_2 \)or \( ar{ ext{ω}}_2 \), components of the imaginary parts produce specific outcomes. The expression \( ext{Re}( ext{ω}_1 ar{ ext{ω}}_2 ) \)has an imaginary term that is part of the given pairs to create products that may have both real and imaginary parts.

• Note that if the imaginary results complement each other, this could lead to simplification or specific acceptable solution conditions.
• Subsequently checking conditions like \( ad + bc = 0 \)helps determine whether parts should cancel or not.

Thus, focusing on the imaginary part helps us dissect the calculations leading to proper real or imaginary results in solutions.

Real Part

The real part of a complex number is simply the actual number without the imaginary unit \( i \). For any complex number \( z = a + ib \), the real part is \( a \). In problems dealing with expressions such as \( z_1 \bar{z}_2 \) and \( ext{ω}_1 ar{ ext{ω}}_2 \), finding the real part involves identifying terms that are free of \( i \).

When analyzing complex expressions, like \( ext{Re}( ext{ω}_1 ar{ ext{ω}}_2 ) = ab + ad \), the assessment hinges on ensuring balance in computation.

• The condition \( ac - bd = 0 \) implies certain constraints aligning both real and imaginary contributions.
• Verifying such conditions helps conclude whether units lie within given solution options.

The real part tells us about the alignment and result vectors via products in complex operations, thus giving insight into the feasibility of proposed solutions.