https://file.aiquickdraw.com/mathbot/file/1727262736486w5kf6zkz.jpg

The problem presented in the image asks to solve the equation involving a complex number and its modulus:

\[
|z - 1 + 3i| = 4
\]

This is a typical problem of finding a complex number \( z \) given that its distance to a point on the complex plane is a fixed value. 

### Step-by-step approach:

1. **Interpreting the expression**: 
   - The modulus \( |z - 1 + 3i| = 4 \) indicates that the distance between \( z \) and the complex number \( 1 - 3i \) is 4.

2. **Geometric interpretation**: 
   - On the complex plane, this represents a circle centered at \( (1, -3) \) with a radius of 4.

3. **Possible solutions**:
   - The point \( z \) can be any point on this circle. There is no unique solution for \( z \), but there are many complex numbers satisfying this condition.

Looking at the provided multiple-choice options, it seems to offer different ways of expressing the answer.

Do you want me to go through the options and clarify the correct interpretation, or should we calculate a particular solution for \( z \)? 

Also, here are some related questions to deepen understanding:

1. What is the modulus of a complex number and how is it calculated?
2. How can the equation \( |z - 1 + 3i| = 4 \) be visualized geometrically?
3. What are the general properties of circles in the complex plane?
4. How can complex numbers be represented in polar form?
5. How does the distance formula in the complex plane compare to the Cartesian plane?

**Tip**: When dealing with complex numbers, visualizing them on the complex plane helps in understanding geometric properties such as distance, which is key in modulus equations.

Solve the equation |z - 1 + 3i| = 4 for complex number z.

Learn how to solve the equation |z - 1 + 3i| = 4 for a complex number z. This problem involves calculating the modulus of a complex number and interpreting it geometrically as the equation of a circle in the complex plane. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation