[THOUGHT] Let me analyze this problem. We have:

• $z_1 = 1 + 2i$
• $z_2 = 3i$
• $S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}$

Let $z = x + iy$. Then:

• $|z - z_1| = |x + iy - 1 - 2i| = |(x-1) + i(y-2)| = \sqrt{(x-1)^2 + (y-2)^2}$
• $|z - z_2| = |x + iy - 3i| = |x + i(y-3)| = \sqrt{x^2 + (y-3)^2}$

The condition is $|z - z_1| = 2|z - z_2|$, which gives us: $\sqrt{(x-1)^2 + (y-2)^2} = 2\sqrt{x^2 + (y-3)^2}$

Squaring both sides: $(x-1)^2 + (y-2)^2 = 4(x^2 + (y-3)^2)$

Let me expand and simplify this to find the equation of the circle. [/THOUGHT] [PYTHON] from sympy import symbols, expand, simplify, sqrt, Eq, solve

x, y = symbols('x y')

Left side: |z - z1|^2 = (x-1)^2 + (y-2)^2

left = (x - 1)**2 + (y - 2)**2

Right side: 4|z - z2|^2 = 4(x^2 + (y-3)^2)

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

Expand both sides

left_expanded = expand(left) right_expanded = expand(right)

print("Left side expanded:", left_expanded) print("Right side expanded:", right_expanded)

Move everything to one side: left - right = 0

equation = expand(left_expanded - right_expanded) print("Equation (left - right = 0):", equation)

Rearrange to standard form: ax^2 + ay^2 + bx + cy + d = 0

Then divide by coefficient of x^2 to get: x^2 + y^2 + (b/a)x + (c/a)y + d/a = 0

Circle form: (x - h)^2 + (y - k)^2 = r^2

where h = -b/(2a), k = -c/(2a), r^2 = h^2 + k^2 - d/a

from sympy import collect, Poly

equation = -3x^2 - 3y^2 - 2x + 20y - 31 = 0

Multiply by -1: 3x^2 + 3y^2 + 2x - 20y + 31 = 0

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

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

Center: (-D/2, -E/2), radius: sqrt(D^2/4 + E^2/4 - F)

From equation: -3x^2 - 3y^2 - 2x + 20y - 31 = 0

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

from sympy import Rational

D = Rational(2, 3) E = Rational(-20, 3) F = Rational(31, 3)

x0 = -D / 2 y0 = -E / 2

print(f"x0 = {x0}") print(f"y0 = {y0}")

r_squared = D2 / 4 + E2 / 4 - F print(f"r^2 = {r_squared}")

r = sqrt(r_squared) print(f"r = {r}")

Calculate 9(x0 + y0)r^2

result = 9 * (x0 + y0) * r_squared print(f"9(x0 + y0)r^2 = {result}") [/PYTHON]

[THOUGHT] The Python calculation gives us:

• Center: $(x_0, y_0) = (-\frac{1}{3}, \frac{10}{3})$
• Radius squared: $r^2 = \frac{8}{9}$
• The value $9(x_0 + y_0)r^2 = 9 \cdot (-\frac{1}{3} + \frac{10}{3}) \cdot \frac{8}{9} = 9 \cdot 3 \cdot \frac{8}{9} = 24$ [/THOUGHT]

\boxed{24}