[THOUGHT] We are given z1 = 1 + 2i, z2 = 3i. The set S is defined as: |x + iy - z1| = 2|x + iy - z2|. This is the equation of a circle (Apollonius circle) because it's a constant ratio of distances to two fixed points.

Let z = x + iy. Then |z - z1| = 2|z - z2|. Square both sides: |z - z1|^2 = 4|z - z2|^2.

Write z = x + iy, z1 = 1 + 2i, z2 = 0 + 3i. Then: |z - z1|^2 = (x - 1)^2 + (y - 2)^2. |z - z2|^2 = (x - 0)^2 + (y - 3)^2.

Equation: (x - 1)^2 + (y - 2)^2 = 4[x^2 + (y - 3)^2].

Expand and bring to standard circle form (x - h)^2 + (y - k)^2 = R^2 to find center (x0, y0) and radius r.

Then compute 9*(x0 + y0)*r^2.

We'll do this with Python symbolic algebra. [/THOUGHT] [PYTHON] import sympy as sp

x, y = sp.symbols('x y')

Define expressions

left = (x - 1)**2 + (y - 2)2 right = 4 * (x2 + (y - 3)**2)

Expand both sides

left_exp = sp.expand(left) right_exp = sp.expand(right)

print("Left expanded:", left_exp) print("Right expanded:", right_exp)

Bring all terms to one side: left - right = 0

eq = left_exp - right_exp print("Equation = 0:", eq)

Simplify eq to standard circle form: x^2 + y^2 + Dx + Ey + F = 0

eq_simplified = sp.simplify(eq) print("Simplified equation:", eq_simplified)

Now group terms: coeff of x^2, y^2, x, y, constant

eq_simplified is -3x**2 - 3y**2 - 2x + 20y - 31 = 0

Divide by -3 to make leading coefficients of x^2 and y^2 equal to 1:

x^2 + y^2 + (2/3)x - (20/3)y + 31/3 = 0

Extract coefficients:

coeff_x2 = eq_simplified.coeff(x, 2) coeff_y2 = eq_simplified.coeff(y, 2) coeff_x = eq_simplified.coeff(x, 1) coeff_y = eq_simplified.coeff(y, 1) const = eq_simplified.as_independent(x, y)[0]

print("Coeff x^2:", coeff_x2) print("Coeff y^2:", coeff_y2) print("Coeff x:", coeff_x) print("Coeff y:", coeff_y) print("Constant:", const)

Since coeff_x2 = coeff_y2 = -3, divide whole equation by -3:

D = coeff_x / coeff_x2 # because after dividing by coeff_x2, coefficient of x is coeff_x / coeff_x2 E = coeff_y / coeff_x2 F = const / coeff_x2

print("D =", D) print("E =", E) print("F =", F)

Circle center: (-D/2, -E/2)

x0 = -D/2 y0 = -E/2 print("Center (x0, y0):", (x0, y0))

Radius squared: r^2 = (D/2)^2 + (E/2)^2 - F

r_sq = (D/2)**2 + (E/2)**2 - F print("r^2 =", r_sq) print("r =", sp.sqrt(r_sq))

Compute 9*(x0 + y0)*r^2

result = 9 * (x0 + y0) * r_sq print("9*(x0 + y0)*r^2 =", result) print("Numerical value:", result.evalf()) [/PYTHON]

[THOUGHT] The Python computation shows that the center is ((-1/3, 10/3)) and (r^2 = 8/9). Then (9(x_0 + y_0) r^2 = 9 \left(-\frac{1}{3} + \frac{10}{3}\right) \cdot \frac{8}{9} = 9 \cdot \frac{9}{3} \cdot \frac{8}{9} = 9 \cdot 3 \cdot \frac{8}{9} = 3 \cdot 8 = 24). The output confirms the value is 24. [/THOUGHT] \boxed{24}