Math Problem Statement
Solution
The problem in the image is asking for the value of the imaginary part of , where is a complex number that satisfies the equation:
Let me walk through the steps to solve for the imaginary part of .
Step-by-step solution:
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Representation of Complex Numbers:
- Let , where is the real part and is the imaginary part of .
- The complex conjugate .
- The magnitude .
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Substitute into the Equation:
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Separate Real and Imaginary Parts:
- The real part of the left-hand side is .
- The imaginary part is .
- The right-hand side has a real part of 12 and an imaginary part of 9.
Therefore, equating real and imaginary parts gives us two equations:
- Real part:
- Imaginary part:
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Solve for :
- From , substitute into the imaginary part equation: Simplifying: Rearranging: Square both sides: Expand the right-hand side: Simplify:
Conclusion:
The imaginary part of is .
Let me know if you need any more details or explanations!
Follow-up questions:
- What is the real part of ?
- Can you explain how the magnitude of a complex number is calculated?
- How does the complex conjugate affect the imaginary part of ?
- What happens if is purely imaginary in the same equation?
- Can you derive the magnitude given the solution for ?
Tip:
When solving complex number problems, always separate real and imaginary parts to form solvable equations.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Magnitude of Complex Numbers
Complex Conjugates
Algebra
Formulas
z = x + yi
\overline{z} = x - yi
|z| = \sqrt{x^2 + y^2}
Separate real and imaginary parts to form equations
Theorems
Properties of Complex Conjugates
Magnitude of Complex Numbers
Suitable Grade Level
Grade 11-12, College Level
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