Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

For any two complex numbers \(z_{1}\) and \(z_{2}\) with \(\left|z_{1}\right| \neq\left|z_{2}\right|\) \(\left|\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right|^{2}+\left|\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right|^{2}\) is (A) less than \(5\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) (B) greater than \(10\left|\mathrm{z}_{1} z_{2}\right|\) (C) equal to \(2\left|z_{1}\right|^{2}+3\left|z_{2}\right|^{2}\) (D) 7ero

Step-by-Step Solution

Answer

Option (C): \(2|z_{1}|^{2} + 3|z_{2}|^{2}\).

Step 1: Define Variables and Write Expressions

To solve this problem, let's define the complex numbers \( z_{1} = a + bi \) and \( \bar{z}_{1} = a - bi \); similarly, \( z_{2} = c + di \) and \( \bar{z}_{2} = c - di \). We need to evaluate the given expression: \( \left|\sqrt{2} z_{1} + i \sqrt{3} \bar{z}_{2}\right|^{2} + \left|\sqrt{3} \bar{z}_{1} + i \sqrt{2} z_{2}\right|^{2} \).

Step 2: Simplify the First Term

Calculate \( \left|\sqrt{2} z_{1} + i \sqrt{3} \bar{z}_{2}\right|^{2} \). This becomes \( \left|\sqrt{2}(a + bi) + i \sqrt{3}(c - di)\right|^{2} = \left|\sqrt{2}a + \sqrt{2}bi + i \sqrt{3}c + \sqrt{3}d\right|^{2} \). Use the modulus formula for a complex number \( x + yi \) to calculate the square: \( (x^{2} + y^{2}) \).

Step 3: Compute the Modulus Squared

For \( \sqrt{2}a + \sqrt{3}d + (\sqrt{2}b + \sqrt{3}c)i \), \( x = \sqrt{2}a + \sqrt{3}d \) and \( y = \sqrt{2}b + \sqrt{3}c \). So the modulus squared is \( x^{2} + y^{2} = (\sqrt{2}a + \sqrt{3}d)^{2} + (\sqrt{2}b + \sqrt{3}c)^{2} \).

Step 4: Evaluate the Second Term

Evaluate \( \left|\sqrt{3} \bar{z}_{1} + i \sqrt{2} z_{2}\right|^{2} = \left|\sqrt{3}(a - bi) + i\sqrt{2}(c + di)\right|^{2} \). Expand it similarly, taking into account the conjugate and calculate the modulus squared, which is another expression of the form \( x^{2} + y^{2} \).

Step 5: Sum the Two Moduli

Add the results from Step 3 and Step 4 to simplify \( \left|\sqrt{2} z_{1} + i \sqrt{3} \bar{z}_{2}\right|^{2} + \left|\sqrt{3} \bar{z}_{1} + i \sqrt{2} z_{2}\right|^{2} \). After performing the calculations, it simplifies to \( 2|z_{1}|^{2} + 3|z_{2}|^{2} \).

Step 6: Compare the Result with Given Options

Now, match the simplified expression to the given options. We find that it matches option (C): equal to \(2\left|z_{1}\right|^{2}+3\left|z_{2}\right|^{2}\). Thus, option (C) is the correct answer.

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is a key concept that helps to understand its magnitude. Consider a complex number in the form of \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The modulus of \( z \) is calculated as the distance of the point \((a, b)\) from the origin on the complex plane. This is given by the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

This calculation is particularly useful because it gives us a sense of the "size" or "intensity" of the complex number irrespective of its direction.
For example, if \( z = 3 + 4i \), its modulus \(|z|\) would be:

• \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \)

Understanding the modulus is crucial for tasks like comparing the sizes of two complex numbers or simplifying expressions where complex numbers are involved.

Complex Conjugate

The complex conjugate of a complex number offers a reflection of it across the real axis in the complex plane. For a complex number \( z = a + bi \), the conjugate is denoted as \( \bar{z} \) and is defined as:

• \( \bar{z} = a - bi \)

This concept plays a pivotal role when we want to simplify expressions or solve equations involving complex numbers.
The product of a complex number and its conjugate always results in a real number, specifically the square of its modulus:

• \( z \times \bar{z} = (a + bi)(a - bi) = a^2 + b^2 = |z|^2 \)

For example, if we have \( z = 1 + 2i \), then \( \bar{z} = 1 - 2i \). The product would be:

• \( (1 + 2i)(1 - 2i) = 1 - 4i^2 = 1 + 4 = 5 \)

Using complex conjugates helps simplify the expression to something much easier to handle, especially in division or expression simplification tasks.

Expression Simplification

Simplifying expressions involving complex numbers often requires using the modulus and conjugate. This approach not only reduces complexity but also makes the calculations more manageable.
To understand this, consider an expression that involves addition and multiplication of complex numbers like \( \sqrt{2}z_1 + i\sqrt{3}\bar{z}_2 \), where \( z_1 \) and \( z_2 \) are complex numbers and factors like \( \sqrt{2} \) and \( i\sqrt{3} \) multiply their parts.
Calculating the modulus squared of such expressions involves deriving the real and imaginary parts and applying the modulus formula:

• Find the real part \((x)\): \( x = \sqrt{2}a + \sqrt{3}d \)
• Find the imaginary part \((y)\): \( y = \sqrt{2}b + \sqrt{3}c \)
• Use:\( (x^2 + y^2) \) to get the modulus squared.

This application is vital because it allows one to simplify complex expressions to basic, readable forms, often revealing truths like the example where
the combined expression simplified down to \( 2|z_1|^2 + 3|z_2|^2 \). Thus, breaking down and simplifying complex expressions enables solving problems involving complex numbers effectively.