Key Concepts and Formulas

• Geometric Interpretation: $|z_1 - z_2|$ represents the distance between complex numbers $z_1$ and $z_2$ in the complex plane. $|z|$ represents the distance of $z$ from the origin.
• Triangle Inequality: For complex numbers $z_a$ and $z_b$, $|z_a + z_b| \leqslant |z_a| + |z_b|$ and $|z_a - z_b| \geqslant ||z_a| - |z_b||$.
• Modulus of a Complex Number: If $z = a + bi$, then $|z| = \sqrt{a^2 + b^2}$.

Step-by-Step Solution

Step 1: Rewrite the Expression and Identify the Fixed Point

We are given the expression $\left|z+\frac{1}{2}(3+4 i)\right|$ and want to find its minimum value, subject to $|z| \leqslant 1$. We can rewrite the expression as $|z - z_0|$, where $z_0 = -\frac{1}{2}(3+4i)$. Therefore, $z_0 = -\frac{3}{2} - 2i$.

Reasoning: Rewriting the expression in the form $|z - z_0|$ highlights that we are finding the distance between the complex number $z$ and a fixed point $z_0$. This is essential for a geometric approach.

Step 2: Calculate the Modulus of the Fixed Point

We calculate $|z_0|$ to determine its distance from the origin: $|z_0| = \left|-\frac{3}{2} - 2i\right| = \sqrt{\left(-\frac{3}{2}\right)^2 + (-2)^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{25}{4}} = \frac{5}{2}$ Thus, $|z_0| = \frac{5}{2}$.

Reasoning: Calculating $|z_0|$ tells us the location of the fixed point relative to the region where $z$ can lie.

Step 3: Analyze the Region for z

We are given that $|z| \leqslant 1$. This means that $z$ lies within or on the unit circle centered at the origin. Since $|z_0| = \frac{5}{2} = 2.5 > 1$, the point $z_0$ lies outside the unit circle.

Reasoning: Since the fixed point $z_0$ is outside the region where $z$ can be, the minimum distance between $z$ and $z_0$ will not be 0. It will occur when $z$ is on the boundary of the region, i.e., when $|z|=1$.

Step 4: Apply the Triangle Inequality

We want to minimize $|z - z_0|$. Using the triangle inequality $|z_a - z_b| \geqslant ||z_a| - |z_b||$, we let $z_a = z$ and $z_b = z_0$: $|z - z_0| \geqslant ||z| - |z_0||$ Substitute $|z_0| = \frac{5}{2}$: $|z - z_0| \geqslant \left||z| - \frac{5}{2}\right|$

Reasoning: The triangle inequality provides a lower bound for the distance $|z - z_0|$. To minimize $|z - z_0|$, we minimize this lower bound, subject to the constraint $|z| \leqslant 1$.

Step 5: Minimize the Lower Bound

We want to minimize $\left||z| - \frac{5}{2}\right|$ subject to $0 \leqslant |z| \leqslant 1$. Let $x = |z|$. Then we want to minimize $\left|x - \frac{5}{2}\right|$ where $0 \leqslant x \leqslant 1$. Since $\frac{5}{2} > 1$, the closest value to $\frac{5}{2}$ in the interval $[0, 1]$ is $x = 1$. Therefore, the minimum value of $\left||z| - \frac{5}{2}\right|$ occurs when $|z| = 1$. Substituting $|z| = 1$, we get: Minimum value $= \left|1 - \frac{5}{2}\right| = \left|-\frac{3}{2}\right| = \frac{3}{2}$.

Reasoning: By considering the range of possible values for $|z|$ and the properties of the absolute value function, we find the value of $|z|$ that minimizes the lower bound.

Step 6: Geometric Interpretation (Verification)

The minimum distance from any point $z$ within or on the unit circle to the fixed point $z_0$ occurs when $z$ is on the line connecting the origin to $z_0$ and $|z| = 1$. The minimum distance is $|z_0| - 1 = \frac{5}{2} - 1 = \frac{3}{2}$.

Reasoning: This geometric argument confirms our algebraic result, providing a clear visual understanding.

Conclusion

The minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is $\frac{3}{2}$.

Common Mistakes & Tips:

• Sign Errors: Be careful with signs when identifying the fixed point $z_0$. The expression $|z + P|$ represents the distance between $z$ and $-P$.
• Triangle Inequality Form: Choose the appropriate form of the triangle inequality based on whether you are finding a maximum or a minimum value. For minimums, $|z_a - z_b| \geqslant ||z_a| - |z_b||$ is usually most effective.
• Geometric Visualization: Always try to visualize complex number problems geometrically. The modulus $|z_1 - z_2|$ is a distance, and the condition $|z| \leqslant R$ means $z$ is inside or on a circle of radius $R$ centered at the origin.

Summary

This problem demonstrates how to combine geometric intuition with the triangle inequality to find the minimum value of an expression involving complex numbers. By identifying the fixed point, its location relative to the allowed region for $z$, and using the appropriate form of the triangle inequality, we systematically determined that the minimum value is $\frac{3}{2}$.

The final answer is \boxed{\frac{3}{2}}, which corresponds to option (C).