[THOUGHT] I need to find the center and radius of the circle defined by the set S, which is the set of all real pairs (x, y) such that the distance from (x, y) to z1 is twice the distance from (x, y) to z2.

First, I will write the equation |x + iy - z1| = 2|x + iy - z2| in terms of x and y. Then, I will simplify it to the standard form of a circle equation (x - x0)^2 + (y - y0)^2 = r^2. Finally, I will extract the center (x0, y0) and radius r, and compute 9(x0 + y0)r^2.

[/THOUGHT] [PYTHON] import sympy as sp

x, y = sp.symbols('x y') z1 = 1 + 2sp.I z2 = 3sp.I

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

equation = sp.Abs(x + ysp.I - z1) - 2sp.Abs(x + y*sp.I - z2)

Square both sides to eliminate absolute values

squared_equation = sp.expand(equation**2)

Simplify the equation to standard circle form

simplified_eq = sp.simplify(squared_equation)

Convert to standard form (x - x0)^2 + (y - y0)^2 = r^2

We can complete the square or use sympy's solve to find center and radius

Alternatively, compare coefficients

Let's expand and collect terms

expanded_eq = sp.expand(simplified_eq) coeff_x2 = expanded_eq.coeff(x2) coeff_y2 = expanded_eq.coeff(y2) coeff_xy = expanded_eq.coeff(x*y) coeff_x = expanded_eq.coeff(x) coeff_y = expanded_eq.coeff(y) constant = expanded_eq.subs(x, 0).subs(y, 0)

Since it's a circle, coeff_x2 and coeff_y2 should be equal and non-zero

The general form is A(x^2 + y^2) + Bx + Cy + D = 0

A = coeff_x2 B = coeff_x C = coeff_y D = constant

Center (x0, y0) = (-B/(2A), -C/(2A))

x0 = -B / (2A) y0 = -C / (2A)

Radius r = sqrt( (B^2 + C^2 - 4AD) / (4A^2) )

r_squared = (B2 + C2 - 4AD) / (4*A**2) r = sp.sqrt(r_squared)

Compute 9(x0 + y0)r^2

result = 9 * (x0 + y0) * r_squared

Simplify the result

result = sp.simplify(result)

print(result) [/PYTHON]