Problem 5 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of solutions of the equation \(z^{2}+|z|^{2}=0\), where \(z \in C\) is (A) one (B) two (C) three (D) infinitely many

Step-by-Step Solution

Answer

(D) infinitely many

Step 1: Understand the problem statement

We need to find the number of solutions for the equation \(z^{2}+|z|^{2}=0\) where \(z\) is a complex number. \(|z|^{2}\) represents the modulus squared of a complex number \(z\).

Step 2: Express \(z\) as a complex number

Consider the complex number \(z = x + yi\), where \(x\) and \(y\) are real numbers and \(i\) is the imaginary unit. Calculate \(|z|^{2} = x^2 + y^2\).

Step 3: Substitute into the equation

Substitute \(z = x + yi\) and \(|z|^{2} = x^2 + y^2\) into the original equation. This gives:\[(x + yi)^2 + x^2 + y^2 = 0\].

Step 4: Expand the expression

Expand \((x + yi)^2\):\[(x + yi)^2 = x^2 + 2xyi - y^2\]. Substitute back into the equation to get:\[x^2 - y^2 + 2xyi + x^2 + y^2 = 0\]. This simplifies to:\[2x^2 + 2xyi = 0\].

Step 5: Separate real and imaginary parts

Separate the equation into real and imaginary parts: \(2x^2 = 0\) and \(2xy = 0\).

Step 6: Solve real part equation

The equation \(2x^2 = 0\) implies that \(x = 0\).

Step 7: Solve imaginary part equation

The equation \(2xy = 0\) is trivially satisfied for \(x = 0\), as it reduces to \(0 \cdot y = 0\). Therefore, \(y\) can be any real number.

Step 8: Conclusion

Since \(x = 0\) and \(y\) can take any real value, the solution \(z = 0 + yi = yi\) implies there are infinitely many solutions. Each value of \(y\) gives a solution.

Key Concepts

Modulus of Complex Numbers

The modulus of a complex number is a key concept in understanding complex numbers. It measures the "size" or "length" of the vector corresponding to a complex number on the complex plane. For a complex number represented as \(z = x + yi\), the modulus \(|z|\) is calculated using the formula:

• \(|z| = \sqrt{x^2 + y^2}\)

However, in the context of our problem, we're dealing with \(|z|^2\), which is simply the modulus squared, given as:

• \(|z|^2 = x^2 + y^2\)

This form avoids the square root, making it quite handy in equations. Here, the modulus squared is used in the equation \(z^2 + |z|^2 = 0\). Understanding the modulus helps simplify and separate components to solve the equation.

Real and Imaginary Parts

Complex numbers are built from two components: the real part and the imaginary part. Given a complex number \(z = x + yi\), \(x\) is the real part, and \(yi\) is the imaginary part. This understanding is vital when solving equations involving complex numbers.

To solve \(z^2 + |z|^2 = 0\), we substitute \(z = x + yi\) into the equation to decompose it into two separate parts:

• Real part: \(2x^2\)
• Imaginary part: \(2xy\)

By splitting into these components, we address each separately, simplifying the process of finding solutions for \(x\) and \(y\). The real and imaginary parts dictate the behavior and properties of complex numbers, offering a tangible way to work through such problems.

Solutions of Equations

Solving equations involving complex numbers involves breaking down and analyzing their real and imaginary parts. In the context of the given equation \(z^2 + |z|^2 = 0\), we separated the equation into:

• \(2x^2 = 0\)
• \(2xy = 0\)

The real part \(2x^2 = 0\) immediately gives \(x = 0\), simplifying the solution process.

Once \(x\) is determined, the imaginary part \(2xy = 0\) is trivially satisfied, meaning \(y\) can be any real number. Therefore, complex numbers of the form \(z = yi\) satisfy the equation, indicating infinitely many solutions.

By isolating variables and recognizing when parts of the equation are trivially satisfied, we can quickly reach conclusions about potential solutions. This is powerful when handling complex numbers in equations, as it highlights infinite possibilities in certain scenarios, shown by \(z = yi\). Understanding this approach strengthens comprehension of more intricate algebraic problems.