Key Concepts and Formulas

• Modulus of a Complex Number: If $z = x + iy$, where $x$ and $y$ are real numbers, then $|z| = \sqrt{x^2 + y^2}$, and $|z|^2 = x^2 + y^2$.
• Completing the Square: The method of completing the square transforms a quadratic expression of the form $ax^2 + bx + c$ into $a(x - h)^2 + k$, where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.
• Centroid of Complex Numbers: The centroid of complex numbers $z_1, z_2, \ldots, z_n$ is given by $z = \frac{z_1 + z_2 + \cdots + z_n}{n}$. The sum $\sum_{k=1}^n |z - z_k|^2$ is minimized when $z$ is the centroid.

Step-by-Step Solution

Step 1: Convert to Cartesian Coordinates

We are given $v = |z|^2 + |z-3|^2 + |z-6i|^2$. To work with this algebraically, we express $z$ in Cartesian form: Let $z = x + iy$, where $x$ and $y$ are real numbers. This allows us to express the modulus in terms of $x$ and $y$.

Now we expand each term:

• $|z|^2 = |x + iy|^2 = x^2 + y^2$
• $|z - 3|^2 = |(x - 3) + iy|^2 = (x - 3)^2 + y^2 = x^2 - 6x + 9 + y^2$
• $|z - 6i|^2 = |x + i(y - 6)|^2 = x^2 + (y - 6)^2 = x^2 + y^2 - 12y + 36$

Step 2: Expand and Simplify the Expression for $v$

Substitute these expanded forms back into the expression for $v$: $v = (x^2 + y^2) + (x^2 - 6x + 9 + y^2) + (x^2 + y^2 - 12y + 36)$ Combine like terms: $v = 3x^2 + 3y^2 - 6x - 12y + 45$

Step 3: Complete the Square to Find the Minimum Value

We complete the square for the $x$ and $y$ terms to find the minimum value. First, factor out the coefficient $3$: $v = 3(x^2 - 2x) + 3(y^2 - 4y) + 45$

Complete the square for $x^2 - 2x$: $x^2 - 2x = (x - 1)^2 - 1$

Complete the square for $y^2 - 4y$: $y^2 - 4y = (y - 2)^2 - 4$

Substitute these back into the expression for $v$: $v = 3((x - 1)^2 - 1) + 3((y - 2)^2 - 4) + 45$ $v = 3(x - 1)^2 - 3 + 3(y - 2)^2 - 12 + 45$ $v = 3(x - 1)^2 + 3(y - 2)^2 + 30$

Step 4: Determine the Minimum Value ($v_0$) and the Complex Number ($z_0$) where it's attained

Since $(x - 1)^2 \ge 0$ and $(y - 2)^2 \ge 0$, the minimum value of $v$ occurs when $x = 1$ and $y = 2$. Thus, $z_0 = x + iy = 1 + 2i$. The minimum value is $v_0 = 3(0) + 3(0) + 30 = 30$.

Step 5: Calculate the Final Expression $\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}$

We have $z_0 = 1 + 2i$ and $v_0 = 30$.

1. Calculate $z_0^2$: $z_0^2 = (1 + 2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4 = -3 + 4i$

2. Calculate $\bar{z}_0^3$: $\bar{z}_0 = 1 - 2i$ $\bar{z}_0^3 = (1 - 2i)^3 = 1 - 6i + 12i^2 - 8i^3 = 1 - 6i - 12 + 8i = -11 + 2i$

3. Calculate $2z_0^2 - \bar{z}_0^3 + 3$: $2z_0^2 - \bar{z}_0^3 + 3 = 2(-3 + 4i) - (-11 + 2i) + 3 = -6 + 8i + 11 - 2i + 3 = 8 + 6i$

4. Calculate $|2z_0^2 - \bar{z}_0^3 + 3|^2$: $|8 + 6i|^2 = 8^2 + 6^2 = 64 + 36 = 100$

5. Calculate $v_0^2$: $v_0^2 = 30^2 = 900$

6. Calculate the final expression: $|2z_0^2 - \bar{z}_0^3 + 3|^2 + v_0^2 = 100 + 900 = 1000$

Common Mistakes & Tips

• Be careful with signs when expanding and simplifying complex expressions. A common mistake is to incorrectly calculate powers of $i$.
• Remember that the minimum value of a sum of squared distances is attained at the centroid of the points.
• Double-check your arithmetic when calculating $z_0^2$ and $\bar{z}_0^3$.

Summary

We converted the complex numbers to Cartesian coordinates, completed the square to find the minimum value $v_0 = 30$, and the point where it occurs $z_0 = 1 + 2i$. Then we computed $|2 z_{0}^{2}-\bar{z}_{0}^{3}+3|^{2}+v_{0}^{2} = 1000$.

The final answer is \boxed{1000}, which corresponds to option (A).