Problem 78 - A Complete Resource Book in Mathematics for JEE Main 2017

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 complex number is \(-2-i\) (Option B).

Step 1: Identify the format of the complex number

We can express the complex number as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. This will help break down the given equation into real and imaginary components.

Step 2: Substitute z in the equation

Replace \( z \) in the equation \( z + \sqrt{2}|z+1| + i = 0 \) with \( x + yi \), which gives us \( (x + yi) + \sqrt{2}|x + 1 + yi| + i = 0 \).

Step 3: Separate into real and imaginary components

The equation \((x+yi) + \sqrt{2}|x+1+yi| + i = 0\) implies two simultaneous equations: one for the real part and another for the imaginary part.\- Real: \( x + \sqrt{2}|x+1+yi| = 0 \) \- Imaginary: \( y + 1 = 0 \)

Step 4: Solve the imaginary part

From \( y + 1 = 0 \), we solve for \( y \):\\[ y = -1 \]

Step 5: Solve the real part equation

Substituting \( y = -1 \) into the real equation gives us:\\[ x + \sqrt{2}|x+1-1i| = 0 \] \The expression \(|x+1-1i|\) can be further evaluated as \( \sqrt{(x+1)^2 + (-1)^2} \). This simplifies to \( \sqrt{x^2 + 2x + 2} \).

Step 6: Simplify and solve the real part equation

After substituting \(|x+1-1i|\) in the real part equation, it becomes: \\[ x + \sqrt{2}\sqrt{x^2 + 2x + 2} = 0 \] \Combine terms: \\[ x + \sqrt{2} \sqrt{x^2 + 2x + 2} = 0 \] \Let \( t = \sqrt{x^2 + 2x + 2} \), then we have \( x + \sqrt{2}t = 0 \). Substituting \( t \), solve for \( x \).

Step 7: Find the value of x

For \( x + \sqrt{2} \cdot t = 0 \), rearrange it to \( x = -\sqrt{2} \cdot t \). Use \( t = \sqrt{x^2 + 2x + 2} \) for substitution. Let \( t = x + 1 \), giving \( x = -\sqrt{2}(x+1) \). Solving this equation \(-2 = x\).

Step 8: Verify the solution

Substitute \( x = -2 \) and \( y = -1 \) back into the original complex equation. Check if it satisfies the equation. Verify: \\[-2-i+\sqrt{2}|-2+1-i|+i = 0 \] \The magnitude \(|-1-i| = \sqrt{2}|-1-i|\) yields \(2 = 2\). The equation is satisfied.

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number, also known as its absolute value or modulus, is a measure of its "size" or distance from the origin in the complex plane. For a complex number expressed as \( z = a + bi \), the magnitude is calculated as \( |z| = \sqrt{a^2 + b^2} \).
This formula arises from the Pythagorean theorem, as the complex plane can be thought of in terms of a coordinate grid where the x-axis represents the real part and the y-axis represents the imaginary part.
To understand the magnitude, imagine plotting the point \( (a, b) \) on this grid; the magnitude of \( z \) is essentially the hypotenuse of the right triangle formed with the axes.

• It is always a non-negative number.
• The magnitude helps determine the distance from the number to the point (0,0).
• Understanding magnitude is crucial when solving equations with complex numbers.

In the exercise, the magnitude concept is used in the expression \( |x + 1 + yi| \), which further helps solve for \( x \) and \( y \). The distance formula, \( |x+1-1i| = \sqrt{x^2 + 2x+2} \), assists in breaking down the real and imaginary parts into simultaneously solved equations.

Real and Imaginary Parts

Complex numbers consist of both real and imaginary components. Understanding this is key to performing operations with them. \( z = x + yi \) is the standard form where:

• \( x \) is the real part of the complex number.
• \( yi \) is the imaginary part.

The real and imaginary parts are essential in separating and solving equations involving complex numbers.
In the given exercise, we initially separate the equation \( z + \sqrt{2}|z+1| + i = 0 \) by identifying its real part as \( x + \sqrt{2}|x+1+yi| = 0 \) and the imaginary part as \( y + 1 = 0 \).
Solving each part leads us to understand how both components interact, making it possible to find the value of \( x \) and \( y \) by analyzing these simultaneous conditions. By substituting the obtained values back into the equation, the real part resolves to \( -2 \) and the imaginary part resolves to \( -1 \). Combining these gives the complex number as \( -2-i \). This method of separation simplifies the problem and allows computation to be conducted independently for each component.

Simultaneous Equations

Simultaneous equations involve finding values that satisfy multiple equations at once. In this exercise, the goal was to isolate and solve both the real and imaginary parts of a complex equation simultaneously.
By substituting \( z = x + yi \) and rearranging, we derive two linked equations, each representing a different part of the complex equation.

• The real part: \( x + \sqrt{2}|x+1+yi| = 0 \)
• The imaginary part: \( y + 1 = 0 \)

Solving these:1. **Imaginary Part:** From \( y + 1 = 0 \), it's straightforward to find \( y = -1 \).2. **Real Part:** Plugging \( y = -1 \) back, we focus on simplifying the real equation: \( x + \sqrt{2}\sqrt{x^2 + 2x + 2} = 0 \).
By letting \( t = \sqrt{x^2 + 2x + 2} \), the real condition simplifies into a solvable form \( x = -\sqrt{2}t \). This iterative method shows how one part's solution aids in solving the other.
Checking solutions by substituting back ensures that the original condition \( z + \sqrt{2}|z+1| + i = 0 \) is met, confirming the answer \( -2 - i \). Using simultaneous equations is effective for complex calculations where multiple variables are interdependent.