[THOUGHT] Set w = x+iy. The locus |w-z1| = 2|w-z2| is an Apollonius circle. I will expand the squared distances, convert to standard circle form, then extract center and radius and compute 9(x0+y0)r^2 using Python to avoid arithmetic mistakes. [/THOUGHT] [PYTHON] import sympy as sp

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

z1 = 1+2i -> (1,2); z2 = 3i -> (0,3)

eq = sp.Eq((x-1)**2 + (y-2)*2, 4((x-0)**2 + (y-3)**2)) expr = sp.expand(eq.lhs - eq.rhs) print("Expanded equation (set = 0):", expr)

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

expr is quadratic; collect terms

D = sp.expand(sp.diff(expr, x).subs({x:0, y:0})) # not reliable

Instead directly collect coefficients

poly = sp.Poly(expr, x, y) coeff_x2 = poly.coeffs()[poly.monoms().index((2,0))] if (2,0) in poly.monoms() else 0 coeff_y2 = poly.coeffs()[poly.monoms().index((0,2))] if (0,2) in poly.monoms() else 0 coeff_x = poly.coeffs()[poly.monoms().index((1,0))] if (1,0) in poly.monoms() else 0 coeff_y = poly.coeffs()[poly.monoms().index((0,1))] if (0,1) in poly.monoms() else 0 const = poly.coeffs()[poly.monoms().index((0,0))] if (0,0) in poly.monoms() else 0

print("Coefficients:", coeff_x2, coeff_y2, coeff_x, coeff_y, const)

Normalize to x^2 + y^2 + Dx + Ey + F = 0

expr_norm = sp.expand(expr/coeff_x2) polyN = sp.Poly(expr_norm, x, y) Dx = polyN.coeff_monomial(x) Ey = polyN.coeff_monomial(y) F = polyN.coeff_monomial(1) print("Normalized:", expr_norm)

x0 = -Dx/2 y0 = -Ey/2 r2 = sp.simplify(x02 + y02 - F) val = sp.simplify(9*(x0+y0)*r2)

print("Center (x0,y0):", sp.simplify(x0), sp.simplify(y0)) print("r^2:", r2) print("9(x0+y0)r^2:", val) print("As integer:", sp.Integer(val)) [/PYTHON]

[THOUGHT] From the computed center ((x_0,y_0)=(-1/3,10/3)) and (r^2=8/9), we get (9(x_0+y_0)r^2=24). \boxed{24}