Problem 64 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The 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 \(-1+3i, -1-i\), which matches none of the given options.

Step 1: Understand the Problem

We are given the equation \(z(\overline{z-2i})=2(2+i)\) and need to find the correct solutions for \(z\). This involves dealing with complex numbers and their conjugates.

Step 2: Rewrite the Equation

Substitute \(\overline{z-2i} \) with the conjugate of \(z-2i\), which is \(\overline{z} - 2i\). Thus the equation becomes \(z(\overline{z} - 2i) = 2(2+i)\).

Step 3: Expand Left Side

Expand \(z(\overline{z} - 2i)\) to get \(z\overline{z} - 2iz\). The equation is now \(z\overline{z} - 2iz = 4 + 2i \).

Step 4: Express z as a Complex Number

Let \(z = a + bi\), where \(a\) and \(b\) are real numbers. Then \(\overline{z} = a - bi\). Substitute these into the equation to get \((a + bi)(a - bi) - 2i(a + bi) = 4 + 2i \).

Step 5: Simplify

Calculate \( (a + bi)(a - bi) \), which simplifies to \(a^2 + b^2\). Therefore, the equation becomes \(a^2 + b^2 - 2i(a + bi) = 4 + 2i\).

Step 6: Separate Real and Imaginary Parts

Separate the equation into real part \(a^2 + b^2 - 2b = 4\) and imaginary part \(-2a = 2\).

Step 7: Solve for Variables

From the imaginary part equation \(-2a = 2\), solve for \(a\). We get \(a = -1\). Substitute this into the real part equation \(a^2 + b^2 - 2b = 4\) to get \((-1)^2 + b^2 - 2b = 4\).

Step 8: Simplify and Solve for b

Simplify the equation: \(1 + b^2 - 2b = 4\) becomes \(b^2 - 2b - 3 = 0\). Solve this quadratic equation to find \(b\).

Step 9: Factorize Quadratic Equation

Factorize the quadratic \(b^2 - 2b - 3 = 0\) as \((b-3)(b+1) = 0\). The solutions are \(b = 3\) or \(b = -1\).

Step 10: Find Solutions for z

Using \(a = -1\), the solutions for \(z\) are \(-1 + 3i\) and \(-1 - i\). Match these to the given options.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, representing equations of the form \(ax^2 + bx + c = 0\). These equations are characterized by their degree, which is two, due to the highest exponent being squared. Solving a quadratic equation typically involves finding the values of \(x\) where the equation holds true.
Common methods for solving quadratic equations include:

• Factoring: Expressing the quadratic in a product form, such as \((x-p)(x-q) = 0\), where the solutions are \(x = p\) and \(x = q\).
• Using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), which gives the roots directly from the coefficients \(a\), \(b\), and \(c\).
• Completing the Square: Rewriting the quadratic in a square form \((x + d)^2 = e\), followed by solving for \(x\).

Quadratic equations are significant because they appear in various areas, such as physics, engineering, and even finance. In the context of complex numbers, quadratic equations may have solutions involving complex numbers if the discriminant \(b^2 - 4ac\) is negative, thus resulting in imaginary roots.

Conjugate of a Complex Number

When dealing with complex numbers, it's crucial to understand the concept of a conjugate. A conjugate of a complex number is derived by changing the sign of the imaginary part. For example, if \(z = a + bi\), then its conjugate \(\overline{z}\) is \(a - bi\).
The relevance of conjugates in mathematics includes:

• Simplifying Expressions: Products involving conjugates, such as \(z\overline{z}\), result in real numbers, specifically \(a^2 + b^2\).
• Division of Complex Numbers: Multiplying the numerator and the denominator by the conjugate of the denominator makes division feasible and simplifies the expression.
• Maintaining Equality: In equations involving complex numbers, where a solution requires equating both real and imaginary parts, conjugates help preserve balance and aid in simplification.

Conjugates play an essential role in finding the solutions to complex equations, as demonstrated in the original exercise by transforming given equations to separate real and imaginary components for straightforward resolution.

Imaginary and Real Parts

A complex number is composed of both real and imaginary parts. For a complex number represented as \(z = a + bi\), \(a\) is the real part and \(bi\) is the imaginary part, where \(b\) is a real coefficient and \(i\) is the imaginary unit defined by \(i^2 = -1\).
This breakdown of complex numbers into components has several practical applications:

• Calculations and Operations: Operations on complex numbers—such as addition, subtraction, multiplication, and division—often require separating the parts for manipulation.
• Equating Components: When solving equations, separating the equation into real and imaginary parts amps up the solving process by treating each part independently.
• Visualization and Application: On the complex plane, the axes represent the real (horizontal) and imaginary (vertical) parts, aiding in visual comprehension and real-world applications like electrical engineering.

Finding solutions to complex equations frequently involves isolating these real and imaginary parts, as illustrated in the exercise where the equation was divided into different components for effective solution derivation.