Math Problem Statement
Solution
Let us solve the problem:
We are tasked with determining the complex numbers such that , , and have the same modulus. Let , where .
Step 1: Represent
The expression becomes: The modulus of is:
Step 2: Represent
The expression becomes: The modulus of is:
Step 3: Represent
The expression becomes: The modulus of is:
Step 4: Equate
From Steps 1 and 2, we equate the moduli: Square both sides: Simplify: Expand both sides: Simplify:
Step 5: Substitute into
With , . Let us check if still satisfies the given modulus condition.
The modulus of becomes:
For and when , the modulus becomes . Hence, we equate:
Step 6: Solve for
Square both sides: Simplify: Thus:
Final Answer:
The complex numbers are:
Would you like a detailed explanation of any specific step? Let me know!
Related Questions:
- What is the geometric interpretation of the modulus condition in this problem?
- How do complex transformations like affect the modulus of ?
- Can this problem be solved using polar coordinates for ?
- How would the solution change if the moduli were proportional rather than equal?
- What are the implications of being real () in this problem?
Tip:
When dealing with modulus conditions in complex numbers, always square the moduli to avoid dealing with square roots directly!
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Modulus of Complex Numbers
Algebraic Equations
Formulas
Modulus of a complex number: |a + bi| = sqrt(a^2 + b^2)
Equating moduli: |z - i| = |iz - i| = |z - iz|
Theorems
Properties of Modulus in Complex Numbers
Equality of Modulus in Complex Numbers
Suitable Grade Level
Undergraduate Mathematics or Advanced High School