Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

The complex number which satisfies the equation \(z+\sqrt{2}|z+1|+i=0\) is (A) \(2-i\) (B) \(-2-i\) (C) \(2+i\) (D) \(-2+i\)

Step-by-Step Solution

Answer

The solution is \(z = -2 - i\), or option (B) \(-2-i\).

Step 1: Understand the Equation

The given equation is: \( z + \sqrt{2}|z+1| + i = 0 \). This is an equation involving a complex number, the absolute value function, and imaginary unit \( i \). We need to solve for the complex number \( z \).

Step 2: Isolate the Complex Number

Rewrite the equation by isolating \( z \): \( z = -\sqrt{2}|z+1| - i \). This step helps us express \( z \) explicitly in terms of \(|z+1|\).

Step 3: Substitute Complex Form

Assume \( z = a + bi \), where \( a \) and \( b \) are real numbers. The equation becomes: \( a + bi = -\sqrt{2}\sqrt{(a+1)^2 + b^2} - i \).

Step 4: Equate Real and Imaginary Parts

The real part gives \( a = -\sqrt{2}\sqrt{(a+1)^2 + b^2} \) and the imaginary part gives \( b = -1 \).

Step 5: Solve for \( b \)

From \( b = -1 \), substitute back into the real part of the equation: \( a = -\sqrt{2}\sqrt{(a+1)^2 + (-1)^2} \).

Step 6: Simplify and Solve for \( a \)

Substituting \( b = -1 \) into the equation for \( a \), we get: \( a = -\sqrt{2}\sqrt{(a+1)^2 + 1} \). Solving this gives \( a = -2 \) after squaring both sides and simplifying.

Step 7: Verify the Solution

Substitute \( a = -2 \) and \( b = -1 \) back into the original equation to verify: \( z = -2 - i \). Check if \( |z+1| \) satisfies \( z + \sqrt{2}|z+1| + i = 0 \). This confirms that the solution satisfies the given equation.

Key Concepts

Absolute Value of Complex Numbers

When dealing with complex numbers, the absolute value (or modulus) plays an important role. The absolute value of a complex number \( z = a + bi \) is denoted as \(|z|\) and calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

This represents the distance of the point \((a, b)\) from the origin on the complex plane. Understanding this concept is crucial when analyzing any equation involving complex numbers, as it helps determine the magnitude irrespective of direction. In the problem we solved, we specifically dealt with \(|z+1|\). Calculating \(|z+1|\) required substituting \( z = a + bi \) into \((a+1)^2 + b^2\) under the square root, demonstrating how the absolute value equation comes into play. This helps in separating equations into more manageable parts, allowing you to simplify and solve them step-by-step.
For complex numbers, the absolute value helps measure the size, much like the concept of length in real numbers. It's an essential tool in connecting algebraic representations to geometric interpretations.

Imaginary Unit

The imaginary unit, denoted by \(i\), is fundamental to complex numbers and extends our number system beyond real numbers. The key property of the imaginary unit is that \(i^2 = -1\). This characteristic is essential in various fields such as engineering, physics, and computer science.
Imaginary numbers allow us to solve equations where the solution involves the square root of a negative number. For instance, the equation \(x^2 + 1 = 0\) has no real solutions because the square of any real number is non-negative. However, in the complex number system, the solution is \(x = i\) and \(x = -i\).
In our problem, the imaginary unit \(i\) appears in the equation itself: \(z + \sqrt{2}|z+1| + i = 0\). It is crucial for separating the complexities of the real and imaginary parts of the equation. By balancing these parts independently, we could isolate \(z\) and find the solution reliably. Imaginary units give us the means to expand our mathematical horizons and comprehend phenomena that are not limited to real number solutions.

Equation Solving in Complex Numbers

Solving equations involving complex numbers can initially seem daunting, but breaking down the process into systematic steps can simplify things tremendously. When given an equation like \(z + \sqrt{2}|z+1| + i = 0\), it is essential to isolate the complex number \(z\) and identify its components.
The strategy involves expressing \(z\) in the form \(a + bi\), separating the equation into real and imaginary parts. For instance, the equation becomes two simpler parts:

• Real part: \(a = -\sqrt{2}\sqrt{(a+1)^2 + b^2}\)
• Imaginary part: \(b = -1\)

Isolating and solving each part separately makes it easier to navigate through complex equations. In this approach, assuming values for \(a\) and \(b\), and checking them against the equation, helps verify their correctness.
For each distinct part of the problem, verify if the assumptions hold when substituted back into the original equation. This confirms the accuracy of the solution, as we found with \(a = -2\) and \(b = -1\) in this exercise. Solving complex equations involves balancing both the magnitudes and angles of complex numbers, akin to solving puzzles where each part fits together logically.