[THOUGHT] The problem asks for the parameters of a circle defined by the equation $|z - z_1| = 2|z - z_2|$, where $z = x + iy$, $z_1 = 1 + 2i$, and $z_2 = 3i$. This is a classic Apollonius circle problem.

Let $z = x + iy$. The condition is $|z - (1 + 2i)| = 2|z - 3i|$. Squaring both sides to remove the modulus, we get: $|z - z_1|^2 = 4|z - z_2|^2$.

Let's expand this in terms of $x$ and $y$: $(x - 1)^2 + (y - 2)^2 = 4[x^2 + (y - 3)^2]$.

I will use Python to expand this equation, rearrange it into the standard circle equation form $(x - x_0)^2 + (y - y_0)^2 = r^2$, and extract the center $(x_0, y_0)$ and radius $r$. Finally, I will calculate $9(x_0 + y_0)r^2$.

Steps:

1. Define the variables $x$ and $y$.
2. Set up the equation $(x - 1)^2 + (y - 2)^2 = 4(x^2 + (y - 3)^2)$.
3. Simplify the equation to find the coefficients of $x^2, y^2, x, y$ and the constant term.
4. Determine the center $(x_0, y_0)$ and radius squared $r^2$.
5. Compute the final value $9(x_0 + y_0)r^2$. [/THOUGHT] [PYTHON] from sympy import symbols, expand, Eq, solve

x, y = symbols('x y', real=True)

Define the points

z1_real, z1_imag = 1, 2 z2_real, z2_imag = 0, 3

Define the squared distances

dist_sq_1 = (x - z1_real)**2 + (y - z1_imag)**2 dist_sq_2 = (x - z2_real)**2 + (y - z2_imag)**2

The equation is |z - z1|^2 = 4 * |z - z2|^2

equation = Eq(dist_sq_1, 4 * dist_sq_2)

Expand the equation

expanded_eq = expand(equation.lhs - equation.rhs) print(f"Expanded equation (lhs - rhs = 0): {expanded_eq}")

The equation should be of the form A(x^2 + y^2) + Dx + Ey + F = 0

We can extract coefficients.

Since it's a circle, coeff of x^2 and y^2 should be the same (let's call it A).

Then x^2 + y^2 + (D/A)x + (E/A)y + (F/A) = 0

Center (x0, y0) = (-D/2A, -E/2A)

Radius squared r^2 = x0^2 + y0^2 - F/A

coeff_x2 = expanded_eq.coeff(x, 2) coeff_y2 = expanded_eq.coeff(y, 2) coeff_x = expanded_eq.coeff(x, 1) coeff_y = expanded_eq.coeff(y, 1) constant = expanded_eq.subs({x: 0, y: 0})

print(f"Coeff x^2: {coeff_x2}") print(f"Coeff y^2: {coeff_y2}") print(f"Coeff x: {coeff_x}") print(f"Coeff y: {coeff_y}") print(f"Constant: {constant}")

A = coeff_x2 D = coeff_x E = coeff_y F = constant

Calculate center and radius

x0 = -D / (2 * A) y0 = -E / (2 * A) r_squared = x02 + y02 - F / A

print(f"Center (x0, y0): ({x0}, {y0})") print(f"Radius squared r^2: {r_squared}")

Calculate the final answer

result = 9 * (x0 + y0) * r_squared print(f"Final result: {result}") [/PYTHON] [THOUGHT] The Python code successfully expanded the equation and calculated the parameters. The expanded equation is $-3x^2 - 2x - 3y^2 + 20y - 31 = 0$. Dividing by -3, we get $x^2 + \frac{2}{3}x + y^2 - \frac{20}{3}y + \frac{31}{3} = 0$. Comparing to $x^2 + y^2 - 2x_0x - 2y_0y + x_0^2 + y_0^2 - r^2 = 0$: $-2x_0 = 2/3 \implies x_0 = -1/3$. $-2y_0 = -20/3 \implies y_0 = 10/3$. $x_0^2 + y_0^2 - r^2 = 31/3$. $(-1/3)^2 + (10/3)^2 - r^2 = 31/3$. $1/9 + 100/9 - r^2 = 31/3$. $101/9 - r^2 = 93/9$. $r^2 = (101 - 93)/9 = 8/9$.

The Python output confirms: Center $(x_0, y_0) = (-1/3, 10/3)$. Radius squared $r^2 = 8/9$. Final result calculation: $9(x_0 + y_0)r^2 = 9(-1/3 + 10/3)(8/9) = 9(3)(8/9) = 24$.

Let me double check the arithmetic manually. Equation: $(x-1)^2 + (y-2)^2 = 4(x^2 + (y-3)^2)$ $x^2 - 2x + 1 + y^2 - 4y + 4 = 4(x^2 + y^2 - 6y + 9)$ $x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36$ Move LHS to RHS: $3x^2 + 3y^2 + 2x - 20y + 31 = 0$ Divide by 3: $x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0$ Complete the square: $(x + \frac{1}{3})^2 - \frac{1}{9} + (y - \frac{10}{3})^2 - \frac{100}{9} + \frac{31}{3} = 0$ $(x + \frac{1}{3})^2 + (y - \frac{10}{3})^2 = \frac{1}{9} + \frac{100}{9} - \frac{93}{9}$ $(x + \frac{1}{3})^2 + (y - \frac{10}{3})^2 = \frac{8}{9}$

So, center $(x_0, y_0) = (-1/3, 10/3)$. Radius squared $r^2 = 8/9$.

We need to find $9(x_0 + y_0)r^2$. $x_0 + y_0 = -1/3 + 10/3 = 9/3 = 3$. $r^2 = 8/9$. Value = $9 * 3 * (8/9) = 3 * 8 = 24$.

The calculation is consistent. [/THOUGHT] \boxed{24}

The equation of the set $S$ is given by $|x + iy - (1 + 2i)| = 2|x + iy - 3i|$. Squaring both sides, we have: $|x - 1 + i(y - 2)|^2 = 4|x + i(y - 3)|^2$ $(x - 1)^2 + (y - 2)^2 = 4(x^2 + (y - 3)^2)$ $x^2 - 2x + 1 + y^2 - 4y + 4 = 4(x^2 + y^2 - 6y + 9)$ $x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36$ Rearranging terms to one side: $3x^2 + 3y^2 + 2x - 20y + 31 = 0$ Dividing by 3 to normalize the coefficients of $x^2$ and $y^2$: $x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0$ Completing the square for $x$ and $y$: $\left(x + \frac{1}{3}\right)^2 - \frac{1}{9} + \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} + \frac{31}{3} = 0$ $\left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 = \frac{1}{9} + \frac{100}{9} - \frac{93}{9}$ $\left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 = \frac{8}{9}$

Thus, the center is $(x_0, y_0) = \left(-\frac{1}{3}, \frac{10}{3}\right)$ and the radius squared is $r^2 = \frac{8}{9}$. We need to find the value of $9(x_0 + y_0)r^2$: $x_0 + y_0 = -\frac{1}{3} + \frac{10}{3} = \frac{9}{3} = 3$ $9(x_0 + y_0)r^2 = 9(3)\left(\frac{8}{9}\right) = 27 \cdot \frac{8}{9} = 3 \cdot 8 = 24$

\boxed{24}