https://file.aiquickdraw.com/mathbot/file/1734889880450bu6git1s.jpg

Let us solve the problem:

We are tasked with determining the complex numbers \( z \) such that \( |z - i| \), \( |iz - i| \), and \( |z - iz| \) have the same modulus. Let \( z = x + yi \), where \( x, y \in \mathbb{R} \).

---

### Step 1: Represent \( |z - i| \)
The expression \( z - i \) becomes:
\[
z - i = (x + yi) - i = x + (y-1)i.
\]
The modulus of \( z - i \) is:
\[
|z - i| = \sqrt{x^2 + (y-1)^2}.
\]

---

### Step 2: Represent \( |iz - i| \)
The expression \( iz - i \) becomes:
\[
iz - i = i(x + yi) - i = -y + xi - i = x + (-y-1)i.
\]
The modulus of \( iz - i \) is:
\[
|iz - i| = \sqrt{x^2 + (-y-1)^2}.
\]

---

### Step 3: Represent \( |z - iz| \)
The expression \( z - iz \) becomes:
\[
z - iz = (x + yi) - i(x + yi) = x + yi - (-y + xi) = x + y + (y - x)i.
\]
The modulus of \( z - iz \) is:
\[
|z - iz| = \sqrt{(x + y)^2 + (y - x)^2}.
\]

---

### Step 4: Equate \( |z - i| = |iz - i| \)
From Steps 1 and 2, we equate the moduli:
\[
\sqrt{x^2 + (y-1)^2} = \sqrt{x^2 + (-y-1)^2}.
\]
Square both sides:
\[
x^2 + (y-1)^2 = x^2 + (-y-1)^2.
\]
Simplify:
\[
(y-1)^2 = (-y-1)^2.
\]
Expand both sides:
\[
y^2 - 2y + 1 = y^2 + 2y + 1.
\]
Simplify:
\[
-2y = 2y \quad \implies \quad y = 0.
\]

---

### Step 5: Substitute \( y = 0 \) into \( |z - iz| \)
With \( y = 0 \), \( z = x + 0i = x \). Let us check if \( |z - iz| \) still satisfies the given modulus condition.

\[
z - iz = x - ix = x(1 - i).
\]
The modulus of \( z - iz \) becomes:
\[
|z - iz| = |x(1 - i)| = |x| \cdot |1 - i| = |x| \cdot \sqrt{2}.
\]

For \( z - i \) and \( iz - i \) when \( y = 0 \), the modulus becomes \( \sqrt{x^2 + 1} \). Hence, we equate:
\[
\sqrt{x^2 + 1} = |x| \cdot \sqrt{2}.
\]

---

### Step 6: Solve for \( x \)
Square both sides:
\[
x^2 + 1 = 2x^2.
\]
Simplify:
\[
x^2 = 1.
\]
Thus:
\[
x = \pm 1.
\]

---

### Final Answer:
The complex numbers \( z \) are:
\[
z = 1 \quad \text{or} \quad z = -1.
\]

Would you like a detailed explanation of any specific step? Let me know!

---

### Related Questions:
1. What is the geometric interpretation of the modulus condition in this problem?
2. How do complex transformations like \( iz \) affect the modulus of \( z \)?
3. Can this problem be solved using polar coordinates for \( z \)?
4. How would the solution change if the moduli were proportional rather than equal?
5. What are the implications of \( z \) being real (\( y = 0 \)) in this problem?

---

### Tip:
When dealing with modulus conditions in complex numbers, always square the moduli to avoid dealing with square roots directly!

Determine complex numbers z such that |z - i|, |iz - i|, and |z - iz| have the same modulus.

Learn how to determine the complex numbers z such that |z - i|, |iz - i|, and |z - iz| have equal modulus. This problem involves key concepts in complex numbers, including modulus properties and solving algebraic equations. A step-by-step solution is provided for advanced high school and undergraduate math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation