Key Concepts and Formulas

• Modulus of a complex number: For $z = x+iy$, its modulus is $|z| = \sqrt{x^2+y^2}$, which represents the distance of the point $(x,y)$ from the origin $(0,0)$ in the complex plane.
• Distance between two complex numbers: The distance between two complex numbers $z_1$ and $z_2$ is given by $|z_1-z_2|$.
• Collinearity of three complex numbers: Three distinct complex numbers $z_a, z_b, z_c$ are collinear if and only if $\frac{z_b-z_a}{z_c-z_a}$ is a real number. Geometrically, this means the vector from $z_a$ to $z_b$ is parallel to the vector from $z_a$ to $z_c$.

Step-by-Step Solution

Step 1: Calculate the distance between $z_1$ and $z_0$.

We are given $z_1 = 1+i$ and $z_0 = \frac{1}{2}(1+3i)$. We first find the complex number $z_1 - z_0$, and then find its modulus to determine the distance between the two points. $z_1 - z_0 = (1+i) - \frac{1}{2}(1+3i) = 1 + i - \frac{1}{2} - \frac{3}{2}i = \frac{1}{2} - \frac{1}{2}i$ Now, we calculate the modulus: $|z_1 - z_0| = \left|\frac{1}{2} - \frac{1}{2}i\right| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$ This tells us the distance between $z_1$ and $z_0$ is $\frac{1}{\sqrt{2}}$.

Step 2: Determine the distance between $z_2$ and $z_0$.

We are given that $|z_1 - z_0||z_2 - z_0| = 1$. We substitute the value of $|z_1 - z_0|$ calculated in Step 1 into this equation and solve for $|z_2 - z_0|$. $\frac{1}{\sqrt{2}}|z_2 - z_0| = 1$ $|z_2 - z_0| = \sqrt{2}$ This tells us the distance between $z_2$ and $z_0$ is $\sqrt{2}$. Since the radius of the circle $C$ is 1, and $\sqrt{2} > 1$, $z_2$ is indeed outside the circle.

Step 3: Apply the collinearity condition to find the relationship between vectors.

Since $z_0, z_1,$ and $z_2$ are collinear, the vector $z_2 - z_0$ is a real scalar multiple of the vector $z_1 - z_0$. Thus, we can write: $z_2 - z_0 = k(z_1 - z_0)$ where $k$ is a real number. Taking the modulus of both sides, we get: $|z_2 - z_0| = |k||z_1 - z_0|$ Substituting the known values, we have: $\sqrt{2} = |k|\left(\frac{1}{\sqrt{2}}\right)$ $|k| = 2$ Since $k$ is real, $k = 2$ or $k = -2$.

Step 4: Calculate the possible values of $z_2$.

Using the relation $z_2 - z_0 = k(z_1 - z_0)$, we have $z_2 = z_0 + k(z_1 - z_0)$.

Case 1: $k=2$ $z_2 = z_0 + 2(z_1 - z_0) = 2z_1 - z_0 = 2(1+i) - \frac{1}{2}(1+3i) = 2+2i - \frac{1}{2} - \frac{3}{2}i = \frac{3}{2} + \frac{1}{2}i$

Case 2: $k=-2$ $z_2 = z_0 - 2(z_1 - z_0) = 3z_0 - 2z_1 = 3\left(\frac{1}{2}(1+3i)\right) - 2(1+i) = \frac{3}{2} + \frac{9}{2}i - 2 - 2i = -\frac{1}{2} + \frac{5}{2}i$

Step 5: Calculate $|z_2|^2$ for each possible value of $z_2$.

For Case 1: $z_2 = \frac{3}{2} + \frac{1}{2}i$ $|z_2|^2 = \left(\frac{3}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{9}{4} + \frac{1}{4} = \frac{10}{4} = \frac{5}{2}$

For Case 2: $z_2 = -\frac{1}{2} + \frac{5}{2}i$ $|z_2|^2 = \left(-\frac{1}{2}\right)^2 + \left(\frac{5}{2}\right)^2 = \frac{1}{4} + \frac{25}{4} = \frac{26}{4} = \frac{13}{2}$

Step 6: Identify the smaller value of $|z_2|^2$.

Comparing the two values calculated in Step 5: $\frac{5}{2}$ and $\frac{13}{2}$. The smaller value is $\frac{5}{2}$.

Common Mistakes & Tips

• Remember to consider both positive and negative values of $k$ when dealing with collinearity.
• Double-check the arithmetic when calculating complex numbers and their moduli.
• Ensure that your final answer satisfies all the given conditions (e.g., $z_2$ being outside the circle).

Summary

We found the distances between $z_1$ and $z_0$, and $z_2$ and $z_0$. Using the collinearity condition, we determined two possible values for $z_2$. Then, we calculated $|z_2|^2$ for each case and identified the smaller value. The smaller value of $|z_2|^2$ is $\frac{5}{2}$.

Final Answer

The final answer is \boxed{\frac{5}{2}}, which corresponds to option (B).