Problem 64 - A Complete Resource Book in Mathematics for JEE Main

Question

vvThe solutions of the equation \(z(\overline{z-2 i})=2(2+i)\) are (A) \(3+i, 3-i\) (B) \(1+3 i, 1-3 i\) (C) \(1+3 i, 1-i\) (D) \(1-3 i, 1+i\)

Step-by-Step Solution

Answer

The solutions are (C) \(1+3i, 1-i\).

Step 1: Understand the equation

The given equation is \(z(\overline{z - 2i}) = 2(2+i)\). Here, \(\overline{z - 2i}\) denotes the complex conjugate of \(z - 2i\). Start by expressing \(z\) as \(x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit.

Step 2: Find the complex conjugate

Express \(z - 2i = (x + yi) - 2i = x + (y-2)i\). The complex conjugate of this is \(\overline{z - 2i} = x - (y-2)i = x - yi + 2i\).

Step 3: Substitute and simplify

Substitute \(z = x + yi\) and \(\overline{z - 2i} = x - yi + 2i\) into the equation: \((x + yi)(x - yi + 2i) = 2(2+i)\). Expand the left side: \(x^2 - xy^2i + 2xi + y^2 + 2yi^2 = x^2 - xy^2i + 2xi - 2y + y^2 \).

Step 4: Equate real and imaginary parts

The product equals \(x^2 + y^2 - 2y + 2xi - xy^2i\). This must equal \(4 + 2i\). Now, equate the real parts: \(x^2 + y^2 - 2y = 4\) and equate the imaginary parts: \(2x = 2\).

Step 5: Solve for x and y

From \(2x = 2\), we get \(x = 1\). Substitute \(x = 1\) into the real part equation: \(1 + y^2 - 2y = 4\), simplify to \(y^2 - 2y - 3 = 0\). Factor the quadratic: \((y - 3)(y + 1) = 0\). Thus, \(y = 3\) or \(y = -1\).

Step 6: Identify the solutions

The solutions for \(z\) are \(z = 1 + 3i\) and \(z = 1 - i\).

Key Concepts

Complex Conjugate

Complex numbers often have a companion known as the **complex conjugate**. It's easy to find. The conjugate of a complex number involves changing the sign of its imaginary part while keeping the real part unchanged. For example, if you have a complex number expressed as \( a + bi \), the complex conjugate will be \( a - bi \).
Understanding complex conjugates is important because they help in simplifying expressions and solving equations.

• In our exercise, we have \( \overline{z - 2i} \), which means to take the conjugate of \( z - 2i \). If \( z = x + yi \), subtracting \( 2i \) would give \( (x + yi) - 2i = x + (y-2)i \).
• Thus, the conjugate \( \overline{z - 2i} = x - (y-2)i = x - yi + 2i \).

This change is crucial for solving the equation, as it helps in separating the real and imaginary parts, facilitating the solution process.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It has the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In our exercise, we arrive at a quadratic equation when solving for the variable \( y \).
The equation \( y^2 - 2y - 3 = 0 \) emerges from equating the real parts of the original equation.

• This specific form is a typical quadratic equation where the variable is \( y \).
• Solving it can involve methods such as factoring, completing the square, or using the quadratic formula.

For simplicity, we use factoring: the factors of \( y^2 - 2y - 3 \) are \((y - 3)(y + 1)\). Thus, setting each factor equal to zero gives two potential solutions: \( y = 3 \) and \( y = -1 \). This process shows how quadratic equations are frequently part of solving complex number problems.

Real and Imaginary Parts

The real and imaginary parts are two important aspects of complex numbers. A complex number like \( z = x + yi \) consists of:

• **Real Part**: The term \( x \) is the real part.
• **Imaginary Part**: The term \( yi \) signifies the imaginary part.

In our problem, we equate real and imaginary parts to solve the equation:

• For the real part: After simplifying the equation, we have \( x^2 + y^2 - 2y = 4 \). This equation contains only real components.
• For the imaginary part: Equating \( 2x \) from \( (x^2 + y^2 - 2y + 2xi) \) to \( 2 \) gives \( 2x = 2 \), leading to \( x = 1 \).

This separation of real and imaginary parts simplifies solving because it converts the equation into simpler, more manageable pieces.

Equation Solving

Solving equations involving complex numbers can seem challenging, but understanding the steps can make it manageable. The essence involves:

• Expressing complex numbers in terms of their real and imaginary parts.
• Using identities such as the complex conjugate.
• Isolating and equating the real and imaginary parts separately.

In our exercise, after setting up the equation \((x + yi)(x - yi + 2i) = 2(2+i)\), we equate the parts:

• Simplifying, we balance the real part \( x^2 + y^2 - 2y = 4 \) and the imaginary part \( 2x = 2 \) independently.
• This gives solutions for \( x \) and a quadratic equation for \( y \).
• Solving these gives the solutions for \( z \) as \( 1 + 3i \) and \( 1 - i \), showing how the individual parts come together to form the complete solution.

The method requires attention to separate complex numbers into their real and imaginary components, allowing systematic solving through familiar algebraic techniques.