[THOUGHT] I will translate the locus condition into an equation in x and y, expand and simplify it to standard circle form, then read off the center and radius. I will use Python (sympy) to handle the algebra and then compute 9(x0 + y0)r^2. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

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

Define the condition: |x+iy - (1+2i)| = 2 |x+iy - 3i|

Corresponding squared distances:

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

Expand and simplify to get circle equation

circle_eq = sp.expand(expr) print("Circle equation:", circle_eq)

Put in standard form x^2 + y^2 + Dx + Ey + F = 0

Then center = (-D/2, -E/2), radius^2 = (D^2 + E^2)/4 - F

D = sp.expand(sp.diff(circle_eq, x, 1).subs({x:0, y:0})) # Coef of x2 E = sp.expand(sp.diff(circle_eq, y, 1).subs({x:0, y:0})) # Coef of y2

Actually easier: collect

circle_collected = sp.collect(circle_eq, [x2, y2, x, y], evaluate=False) print("Collected:", circle_collected)

Manually read coefficients: since equation is of form ax^2+by^2+cx+dy+e=0

a = sp.expand(sp.diff(circle_eq, x, 2))/2 # coefficient of x^2 b = sp.expand(sp.diff(circle_eq, y, 2))/2 # coefficient of y^2

Alternatively, use sp.Poly

poly = sp.Poly(circle_eq, x, y) coeffs = poly.coeffs() monoms = poly.monoms() print("Monomials and coefficients:", list(zip(monoms, coeffs)))

Extract coefficients directly

A = poly.coeffs()[monoms.index((2,0))] # x^2 B = poly.coeffs()[monoms.index((0,2))] # y^2 C = poly.coeffs()[monoms.index((1,0))] # x Dcoef = poly.coeffs()[monoms.index((0,1))] # y Econst = poly.coeffs()[monoms.index((0,0))] # constant term

print("A,B,C,D,E:", A, B, C, Dcoef, Econst)

Standard form requires A = B = 1 (circle). Our equation likely has factor, so divide by A

circle_eq_norm = sp.simplify(circle_eq / A) print("Normalized circle eq:", circle_eq_norm)

poly_norm = sp.Poly(circle_eq_norm, x, y) A_n = poly_norm.coeffs()[poly_norm.monoms().index((2,0))] B_n = poly_norm.coeffs()[poly_norm.monoms().index((0,2))] C_n = poly_norm.coeffs()[poly_norm.monoms().index((1,0))] D_n = poly_norm.coeffs()[poly_norm.monoms().index((0,1))] E_n = poly_norm.coeffs()[poly_norm.monoms().index((0,0))] print("Normalized A,B,C,D,E:", A_n, B_n, C_n, D_n, E_n)

Center and radius from normalized equation: x^2 + y^2 + C_n x + D_n y + E_n = 0

x0 = -C_n/2 y0 = -D_n/2 r2 = (C_n2 + D_n2)/4 - E_n

print("x0, y0, r^2:", x0, y0, r2)

result = 9*(x0 + y0)*r2 print("Result 9(x0 + y0)r^2:", sp.simplify(result)) [/PYTHON]

[THOUGHT] Using the validated Python results, the circle’s center is ((x_0, y_0) = \left(-\frac{1}{3}, \frac{10}{3}\right)) and (r^2 = \frac{8}{9}). The computed value of (9(x_0 + y_0)r^2) is 24. \boxed{24}