Problem Analysis #14

The set $S$ is defined by the equation $|x + iy - z_1| = 2|x + iy - z_2|$, where $z_1 = 1 + 2i$ and $z_2 = 3i$. This represents a circle in the complex plane.

Representing a point as $z = x + iy$, the equation becomes: [ |x - (1 + 2i)| = 2|x - 3i| ]

Squaring both sides to eliminate the absolute values: [ (x - 1)^2 + (y - 2)^2 = 4 \left[ (x - 0)^2 + (y - 3)^2 \right] ]

Expanding both sides: [ x^2 - 2x + 1 + y^2 - 4y + 4 = 4(x^2 + y^2 - 6y + 9) ]

Bringing all terms to one side: [ x^2 + y^2 - 2x - 4y + 5 - 4x^2 - 4y^2 + 24y - 36 = 0 ] [ -3x^2 - 3y^2 - 2x + 20y - 31 = 0 ]

Multiplying by $-1$ to simplify: [ 3x^2 + 3y^2 + 2x - 20y + 31 = 0 ]

Dividing by 3: [ x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0 ]

Completing the square:

• For $x$: $x^2 + \frac{2}{3}x = \left(x + \frac{1}{3}\right)^2 - \left(\frac{1}{3}\right)^2$
• For $y$: $y^2 - \frac{20}{3}y = \left(y - \frac{10}{3}\right)^2 - \left(\frac{10}{3}\right)^2$

Substituting: [ \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} + \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} + \frac{31}{3} = 0 ]

Combining constants: [ -\frac{1}{9} - \frac{100}{9} + \frac{31}{3} = -\frac{101}{9} + \frac{93}{9} = -\frac{8}{9} ]

So: [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 = \frac{8}{9} ]

The center is $(x_0, y_0) = \left(-\frac{1}{3}, \frac{10}{3}\right)$ and the radius squared is $r^2 = \frac{8}{9}$.

Now compute: [ x_0 + y_0 = -\frac{1}{3} + \frac{10}{3} = \frac{9}{3} = 3 ] [ 9(x_0 + y_0) r^2 = 9 \times 3 \times \frac{8}{9} = 3 \times 8 = 24 ]

Thus, the value is 24.

\boxed{24}

[THOUGHT] We will use Python to compute the final value from the algebraic results. [/THOUGHT] [PYTHON] x0 = -1/3 y0 = 10/3 r2 = 8/9 result = 9 * (x0 + y0) * r2 print(result) [/PYTHON]

[THOUGHT] The Python computation confirms that 9(x0 + y0)r^2 equals 24.0, which is equivalent to the integer 24. [/THOUGHT]

\boxed{24}