[THOUGHT] To solve the problem, we start by interpreting the given condition geometrically. The set ( S ) represents the locus of points ((x, y)) such that the distance to ( z_1 = 1 + 2i ) (which corresponds to the point ((1, 2))) is twice the distance to ( z_2 = 3i ) (which corresponds to the point ((0, 3))). This is the definition of an Apollonius circle.

We set up the equation for the distances: [ \sqrt{(x - 1)^2 + (y - 2)^2} = 2 \sqrt{x^2 + (y - 3)^2} ] Squaring both sides to eliminate the square roots, we get: [ (x - 1)^2 + (y - 2)^2 = 4(x^2 + (y - 3)^2) ] Expanding and simplifying this equation, we obtain: [ x^2 - 2x + 1 + y^2 - 4y + 4 = 4x^2 + 4y^2 - 24y + 36 ] [ x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36 ] [ -3x^2 - 3y^2 - 2x + 20y - 31 = 0 ] Multiplying through by (-1) to simplify: [ 3x^2 + 3y^2 + 2x - 20y + 31 = 0 ] Dividing the entire equation by 3: [ x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0 ] Next, we complete the square for (x) and (y): [ x^2 + \frac{2}{3}x + y^2 - \frac{20}{3}y = -\frac{31}{3} ] For (x): [ x^2 + \frac{2}{3}x = \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} ] For (y): [ y^2 - \frac{20}{3}y = \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} ] Substituting these back into the equation: [ \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} + \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} = -\frac{31}{3} ] Combining constants: [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 - \frac{101}{9} = -\frac{31}{3} ] [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 = \frac{8}{9} ] This is the equation of a circle with center (\left(-\frac{1}{3}, \frac{10}{3}\right)) and radius (\frac{2\sqrt{2}}{3}).

Finally, we calculate (9(x_0 + y_0)r^2): [ x_0 + y_0 = -\frac{1}{3} + \frac{10}{3} = 3 ] [ r^2 = \left(\frac{2\sqrt{2}}{3}\right)^2 = \frac{8}{9} ] [ 9(x_0 + y_0)r^2 = 9 \times 3 \times \frac{8}{9} = 24 ]

[PYTHON] import sympy as sp

Define variables

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

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

z1 = sp.Matrix([1, 2]) z2 = sp.Matrix([0, 3]) point = sp.Matrix([x, y]) equation = sp.Eq(sp.sqrt((point - z1).dot(point - z1)), 2 * sp.sqrt((point - z2).dot(point - z2)))

Square both sides to eliminate the square roots

equation_squared = sp.simplify(sp.Eq((point - z1).dot(point - z1), 4 * (point - z2).dot(point - z2)))

Expand and simplify the equation

equation_expanded = sp.expand(equation_squared)

Collect terms to form the standard circle equation

lhs = sp.collect(equation_expanded.lhs - equation_expanded.rhs, [x, y])

Completing the square

x_coeff = lhs.coeff(x) y_coeff = lhs.coeff(y) constant_term = lhs - x_coeffx - y_coeffy

Completing the square for x and y

x_term = sp.complete_the_square(x_coeff, x) y_term = sp.complete_the_square(y_coeff, y)

Extract center and radius

center_x = -x_coeff / (2 * lhs.coeff(x, 2)) center_y = -y_coeff / (2 * lhs.coeff(y, 2)) radius_squared = (center_x2 + center_y2 - constant_term / lhs.coeff(x, 2)).simplify()

Calculate the required value

result = 9 * (center_x + center_y) * radius_squared print(result) [/PYTHON]

[THOUGHT] The error occurred because SymPy does not have a complete_the_square function. However, the manual derivation correctly identified the center (\left(-\frac{1}{3}, \frac{10}{3}\right)) and radius (\frac{2\sqrt{2}}{3}). Using these values, (9(x_0 + y_0)r^2 = 24). [/THOUGHT] \boxed{24}