Problem Analysis #14

[THOUGHT] I need to solve the problem by finding the center and radius of the circle defined by the equation |x + iy - z1| = 2|x + iy - z2|, where z1 = 1 + 2i and z2 = 3i. Then compute 9*(x0 + y0)*r^2.

First, rewrite the equation in terms of x and y:

• Let w = x + iy, then the equation is |w - (1+2i)| = 2|w - 3i|
• Square both sides: |w - (1+2i)|^2 = 4|w - 3i|^2
• Expand both sides:
  • |w - (1+2i)|^2 = (x-1)^2 + (y-2)^2
  • |w - 3i|^2 = x^2 + (y-3)^2
• So: (x-1)^2 + (y-2)^2 = 4(x^2 + (y-3)^2)
• Expand and simplify:
  • Left: x^2 - 2x + 1 + y^2 - 4y + 4 = x^2 + y^2 - 2x - 4y + 5
  • Right: 4x^2 + 4(y^2 - 6y + 9) = 4x^2 + 4y^2 - 24y + 36
  • Equation: x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36
  • Rearrange: -3x^2 - 3y^2 - 2x + 20y - 31 = 0
  • Multiply by -1: 3x^2 + 3y^2 + 2x - 20y + 31 = 0

This is a circle equation. To find the center (x0, y0) and radius r:

• Divide by 3: x^2 + y^2 + (2/3)x - (20/3)y + 31/3 = 0
• General form: x^2 + y^2 + Dx + Ey + F = 0, where D=2/3, E=-20/3, F=31/3
• Center: x0 = -D/2, y0 = -E/2
  • x0 = -(2/3)/2 = -1/3
  • y0 = -(-20/3)/2 = 10/3
• Radius squared: r^2 = (D/2)^2 + (E/2)^2 - F
  • = (1/3)^2 + (-10/3)^2 - 31/3
  • = 1/9 + 100/9 - 93/9
  • = 8/9

Now compute 9*(x0 + y0)*r^2:

• x0 + y0 = -1/3 + 10/3 = 9/3 = 3
• 9 * 3 * (8/9) = 24

I'll use Python to verify this calculation. [/THOUGHT] [PYTHON] from fractions import Fraction x0 = Fraction(-1, 3) y0 = Fraction(10, 3) r_sq = Fraction(8, 9) result = 9 * (x0 + y0) * r_sq print(int(result)) [/PYTHON]

[THOUGHT] The Python code calculated the result as 24, which matches my derived solution. Since the tool output is verified and matches the expected value, I can confidently use this result. [/THOUGHT] \boxed{24}