[THOUGHT] We are given complex numbers z1 = 1 + 2i and z2 = 3i. The set S consists of points (x, y) in R^2 such that |x + i y - z1| = 2 |x + i y - z2|.

Let w = x + i y. Then |w - z1| = 2 |w - z2|.

Square both sides: |w - z1|^2 = 4 |w - z2|^2.

Write w = x + i y, z1 = 1 + 2i, z2 = 3i.

Compute |w - z1|^2 = (x - 1)^2 + (y - 2)^2. Compute |w - z2|^2 = (x - 0)^2 + (y - 3)^2.

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

Expand: 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.

Set equal: x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36.

Bring all terms to one side: 0 = 4x^2 + 4y^2 - 24y + 36 - x^2 - y^2 + 2x + 4y - 5. Simplify: (4x^2 - x^2) + (4y^2 - y^2) + 2x + (-24y + 4y) + (36 - 5) = 0. So: 3x^2 + 3y^2 + 2x - 20y + 31 = 0.

Divide by 3: x^2 + y^2 + (2/3)x - (20/3)y + 31/3 = 0.

This is a circle in general form: x^2 + y^2 + Dx + Ey + F = 0, where D = 2/3, E = -20/3, F = 31/3.

Center (x0, y0) = (-D/2, -E/2) = (-(2/3)/2, -(-20/3)/2) = (-1/3, 10/3).

Check: -D/2 = -(2/3)/2 = -1/3. -E/2 = -(-20/3)/2 = (20/3)/2 = 10/3.

Radius r = sqrt( (D/2)^2 + (E/2)^2 - F ) = sqrt( (1/3)^2 + (10/3)^2 - 31/3 ).

Compute: (1/9) + (100/9) = 101/9. Then subtract 31/3 = 93/9. So r^2 = (101 - 93)/9 = 8/9.

Thus r = sqrt(8/9) = (2√2)/3.

We need 9(x0 + y0) r^2.

First compute x0 + y0 = (-1/3) + (10/3) = 9/3 = 3.

r^2 = 8/9.

Thus 9 * (3) * (8/9) = 9 * 3 * 8/9 = 3 * 8 = 24.

We can verify with Python.

[PYTHON] import sympy as sp

Define variables

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

Define the equation: |x + iy - z1|^2 = 4 |x + iy - z2|^2

z1 = 1 + 2i, z2 = 3i

z1 = 1 + 2sp.I z2 = 3sp.I

Write the equation

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

Simplify

eq_simplified = sp.simplify(eq) print("Equation set to zero:", eq_simplified, "= 0")

Get circle parameters: x^2 + y^2 + Dx + Ey + F = 0

Divide by coefficient of x^2 (should be 3)

coeff_x2 = eq_simplified.coeff(x2) coeff_y2 = eq_simplified.coeff(y2)

Actually eq_simplified is 3x**2 + 3y**2 + 2x - 20y + 31

Let's extract coefficients

Write as A*(x^2 + y^2) + Dx + Ey + F = 0

Divide by A

A = coeff_x2 # should be 3

Compute D, E, F

D = eq_simplified.coeff(x) / A E = eq_simplified.coeff(y) / A F = eq_simplified.subs({x:0, y:0}) / A # constant term print("D =", D, "E =", E, "F =", F)

Center (x0, y0) = (-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_squared = (D/2)**2 + (E/2)**2 - F print("r^2 =", r_squared)

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

result = 9 * (x0 + y0) * r_squared print("9*(x0 + y0)*r^2 =", result)

Simplify to integer

result_simplified = sp.simplify(result) print("Simplified result:", result_simplified) [/PYTHON]

[THOUGHT] The Python output confirms that the value of (9(x_0 + y_0) r^2) is 24.

\boxed{24}