Math Problem Statement

Let z be a complex number that satisfies the equation: \overline{z} + i|z| = 12 + 9i. What is the value of the imaginary part of z?

Solution

The problem in the image is asking for the value of the imaginary part of zz, where zz is a complex number that satisfies the equation:

z+iz=12+9i\overline{z} + i|z| = 12 + 9i

Let me walk through the steps to solve for the imaginary part of zz.

Step-by-step solution:

  1. Representation of Complex Numbers:

    • Let z=x+yiz = x + yi, where xx is the real part and yy is the imaginary part of zz.
    • The complex conjugate z=xyi\overline{z} = x - yi.
    • The magnitude z=x2+y2|z| = \sqrt{x^2 + y^2}.
  2. Substitute into the Equation: (xyi)+ix2+y2=12+9i(x - yi) + i \sqrt{x^2 + y^2} = 12 + 9i

  3. Separate Real and Imaginary Parts:

    • The real part of the left-hand side is xx.
    • The imaginary part is y+x2+y2-y + \sqrt{x^2 + y^2}.
    • 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: x=12x = 12
    • Imaginary part: y+x2+y2=9-y + \sqrt{x^2 + y^2} = 9
  4. Solve for yy:

    • From x=12x = 12, substitute into the imaginary part equation: y+122+y2=9-y + \sqrt{12^2 + y^2} = 9 Simplifying: y+144+y2=9-y + \sqrt{144 + y^2} = 9 Rearranging: 144+y2=y+9\sqrt{144 + y^2} = y + 9 Square both sides: 144+y2=(y+9)2144 + y^2 = (y + 9)^2 Expand the right-hand side: 144+y2=y2+18y+81144 + y^2 = y^2 + 18y + 81 Simplify: 144=18y+81144 = 18y + 81 63=18y63 = 18y y=6318=3.5y = \frac{63}{18} = 3.5

Conclusion:

The imaginary part of zz is 3.5\boxed{3.5}.

Let me know if you need any more details or explanations!


Follow-up questions:

  1. What is the real part of zz?
  2. Can you explain how the magnitude of a complex number is calculated?
  3. How does the complex conjugate affect the imaginary part of zz?
  4. What happens if zz is purely imaginary in the same equation?
  5. Can you derive the magnitude z|z| given the solution for yy?

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