To solve the problem, we begin by interpreting the given condition:

where $z_1 = 1 + 2i$ and $z_2 = 3i$. This is equivalent to:

Step 1: Square Both Sides

Squaring both sides to eliminate the square roots:

Expanding both sides:

• Left-hand side:

  $$(x - 1)^2 + (y - 2)^2 = x^2 - 2x + 1 + y^2 - 4y + 4 = x^2 + y^2 - 2x - 4y + 5$$

• Right-hand side:

  $$4(x^2 + (y - 3)^2) = 4(x^2 + y^2 - 6y + 9) = 4x^2 + 4y^2 - 24y + 36$$

Now equating both sides:

Bring all terms to one side:

Simplify:

Multiply through by $-1$ for simplicity:

Step 2: Complete the Square

Divide the entire equation by 3 to normalize the coefficients of $x^2$ and $y^2$:

Group and complete the square:

• For $x$: x^2 + \frac{2}{3}x = \left(x + \frac{1}{3}\)^2 - \frac{1}{9}
• For $y$: y^2 - \frac{20}{3}y = \left(y - \frac{10}{3}\)^2 - \frac{100}{9}

So the equation becomes:

Combine constants:

Convert $\frac{31}{3}$ to ninths: $\frac{93}{9}$

Thus, the circle has:

• Center: $\left(-\frac{1}{3}, \frac{10}{3}\right)$
• Radius squared: $r^2 = \frac{8}{9}$

Step 3: Compute the Desired Expression

We are asked to compute:

Where:

• $x_0 = -\frac{1}{3}$
• $y_0 = \frac{10}{3}$
• $r^2 = \frac{8}{9}$

Compute:

Final Answer

[THOUGHT] I need to compute the value of $9(x_0 + y_0)r^2$ where $(x_0, y_0)$ is the center and $r$ is the radius of the circle defined by the equation $|x + iy - z_1| = 2|x + iy - z_2|$, with $z_1 = 1 + 2i$ and $z_2 = 3i$.

Let me verify the calculations using Python to ensure accuracy: </think>

[THOUGHT] The problem involves finding the value of $9(x_0 + y_0)r^2$ where the circle $S$ is defined by $|x + iy - z_1| = 2|x + iy - z_2|$, with $z_1 = 1 + 2i$ and $z_2 = 3i$. To solve this, I need to derive the circle equation from the given condition, find its center $(x_0, y_0)$ and radius $r$, and compute the required expression. Let's use Python to perform the symbolic calculations and verify each step. </THOUGHT> [PYTHON] import sympy as sp

Define variables

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

Define z1 and z2

z1 = 1 + 2j z2 = 0 + 3j

The condition is |x + iy - z1| = 2|x + i*y - z2|

Compute squared magnitudes to avoid square roots

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

Expand and simplify to get the circle equation

equation = sp.expand(left - right) circle_eq = sp.Eq(equation, 0)

Convert to standard circle form by completing the square

Rearrange equation to the form x^2 + y^2 + Dx + Ey + F = 0

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

Extract coefficients for x^2, y^2, x, y, constant

coeff_x2 = A.coeff(x2) coeff_y2 = A.coeff(y2) coeff_x = A.coeff(x) coeff_y = A.coeff(y) constant = A.coeff(1)

print(f"Coefficients: x^2: {coeff_x2}, y^2: {coeff_y2}, x: {coeff_x}, y: {coeff_y}, constant: {constant}")

Since coefficients of x^2 and y^2 are equal (both -3), divide by -3 to make them 1

if coeff_x2 == -3 and coeff_y2 == -3: A = sp.expand(A / -3) print("After dividing by -3:", A) else: print("Unexpected coefficients for x^2 or y^2")

Now complete the square for x and y terms

The equation is x^2 + y^2 + (2/3)x - (20/3)y + 31/3 = 0 (after dividing by -3)

But let's use the symbolic expression to complete the square

For x: coefficient of x is (2/3), so (x + 1/3)^2 = x^2 + (2/3)x + 1/9

For y: coefficient of y is -(20/3), so (y - 10/3)^2 = y^2 - (20/3)y + 100/9

Adjust the constant term

x_center = -1/3 y_center = 10/3 r_squared = 8/9

Compute 9*(x0 + y0)*r^2

result = 9 * (x_center + y_center) * r_squared print(f"Center: ({x_center}, {y_center})") print(f"r^2: {r_squared}") print(f"9*(x0 + y0)*r^2 = {result}")

Output the result

print("\nFinal answer:", result) [/PYTHON]

[THOUGHT]

Let's carefully analyze the problem. We are given a condition on real numbers $x$ and $y$:

with $z_1 = 1 + 2i$ and $z_2 = 3i$.

This is the equation of a circle in the complex plane (or the $xy$-plane), defined as the set of points $(x, y)$ whose distance to $z_1$ is twice the distance to $z_2$.

Step 1: Square Both Sides

We square both sides to eliminate the square root:

Step 2: Expand and Simplify

Left-hand side:

Right-hand side:

Set equal:

Bring all terms to one side:

Divide through by $-3$ to simplify:

--- of completing the square:

• For $x$: $\left(x + \frac{1}{3}\right)^2 = x^2 + \frac{2}{3}x + \frac{1}{9}$
• For $y$: $\left(y - \frac{10}{3}\right)^2 = y^2 - \frac{20}{3}y + \frac{100}{9}$

So the equation becomes:

Step 3: Identify Circle Parameters

From the standard form of a circle $(x - x_0)^2 + (y - y_0)^2 = r^2$, we identify:

• Center: $\left(-\frac{1}{3}, \frac{10}{3}\right)$
• Radius squared: $r^2 = \frac{8}{9}$

Step 4: Compute the Desired Expression

We are asked to compute:

Compute:

Final Answer