Question

Find real \(x\) and \(y\) for which \(|z+3|=1-i z,\) where \(z=x+i y\).

Short answer

The solutions are conditions under constraints; in general, the equation simplifies to manifold covering related closest y-values.

Step-by-step solution

Rewrite the equation with real and imaginary parts

Since \(z = x + i y\), substitute \(z\) into the equation: \(|x + i y + 3| = 1 - i (x + i y)\). This will give: \(|(x + 3) + i y| = 1 - ix + y\).

Find the magnitude of the complex number

The magnitude (modulus) of a complex number \(a + ib\) is given by \(\sqrt{a^2 + b^2}\). So, the magnitude \(|(x + 3) + i y|\) becomes: \(\sqrt{(x+ 3)^2 + y^2}\).

Equate the magnitude to the right-hand side

Set the magnitude equal to the real part of the right-hand side equation: \(\sqrt{(x + 3)^2 + y^2} = 1 - y\).

Square both sides of the equation

To eliminate the square root, square both sides: \((x + 3)^2 + y^2 = (1 - y)^2\).

Expand both sides

Expand \((x + 3)^2\) and \((1 - y)^2\): \((x + 3)^2 = x^2 + 6x + 9\) and \((1 - y)^2 = 1 - 2y + y^2\). Insert these back into the equation: \(x^2 + 6x + 9 + y^2 = 1 - 2y + y^2\).

Simplify and solve for x and y

Subtract \(y^2\) from both sides: \(x^2 + 6x + 9 = 1 - 2y\). Move all terms to the left side: \(x^2 + 6x + 9 - 1 + 2y = 0\), or \(x^2 + 6x + 2y + 8 = 0\). Separate variable terms: \(x^2 + 6x = 2 - 2y\) or \((x^2 + 6x + 9 = 2 - 2y - 7)\). Since this equation (i.e., the components) needs to be true for both real and complex parts.

Extract specific conditions

Separate and isolate specific terms by analyzing components. This might include checking symmetries or single occurrences increment.

Key concepts

Complex Number Equations

To solve equations involving complex numbers, it's crucial to understand how to manipulate and work with expressions that combine both real and imaginary parts.
For example, when you're asked to solve for a complex number in an equation, you must often separate the equation into its real and imaginary components.
This process helps transform the complex problem into simpler, more manageable parts.
Complex number equations require both algebraic and geometric reasoning, and practice is key to mastering these skills.

Solving Step-by-Step

When tackling complex number equations, a step-by-step approach is useful.
Breaking down the problem systematically helps ensure you don't miss any important steps or make errors.
Let's go through the steps in solving the given problem:

• Rewrite the equation to separate real and imaginary parts.
• Find the modulus (magnitude) of the complex number involved.
• Set this modulus equal to the right-hand side of the equation.
• Square both sides to eliminate the square root.
• Expand and simplify the resulting equation.
• Isolate and solve for the individual variables (real and imaginary parts).

Approaching the problem in these steps ensures clarity and accuracy.

Real and Imaginary Parts

A complex number has both real and imaginary parts.
In the problem, we denote the complex number as \(z = x + i y\), where \(x\) is the real part and \(y\) is the imaginary part.
When solving complex equations, you often have to separate these parts to simplify the problem.
For instance, from the equation \(\vert x + iy + 3 \vert = 1 - i(x + iy)\), we separate the real part and the imaginary part and handle them individually.
This separation allows you to solve each part independently, simplifying the overall solution process.

Modulus of Complex Numbers

The modulus (or magnitude) of a complex number is a measure of its size, similar to how absolute value measures the distance from zero for real numbers.
For a complex number \(a + ib\), the modulus is calculated as \(\sqrt{a^2 + b^2}\).
In our example, we found the modulus of \( (x + 3) + iy \), which is \( \sqrt{(x+3)^2 + y^2} \).
This value needs to be set equal to the expression given on the right side of the equation.
Finding the modulus and setting it equal to other expressions allows us to break down the problem further and find solutions for the variables involved.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions.
In the given problem, once we have the modulus set equal to the right-hand side, the next step is algebraic.
We square both sides to get rid of the square root, leading to \( (x+3)^2 + y^2 = (1 - y)^2 \).
After expanding and simplifying, we solve for the variables using regular algebraic techniques, like combining like terms and isolating variables.
This step demands accuracy, as small mistakes can lead to incorrect solutions.
Thus, being careful with algebraic manipulation is critical in solving complex number equations efficiently and correctly.