Key Concepts and Formulas

• Modulus as Distance: For complex numbers $z_1$ and $z_2$, $|z_1 - z_2|$ represents the distance between the points corresponding to $z_1$ and $z_2$ in the complex plane.
• Triangle Inequality: For complex numbers $z$, $z_1$, and $z_2$, $|z - z_1| + |z - z_2| \ge |z_1 - z_2|$. The minimum value of $|z - z_1| + |z - z_2|$ is $|z_1 - z_2|$, which occurs when $z$ lies on the line segment connecting $z_1$ and $z_2$.
• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

Step 1: Identify the complex numbers as points in the complex plane.

We are given the expression $|z - 3\sqrt{2}| + |z - p\sqrt{2}i|$. This represents the sum of distances from a point $z$ in the complex plane to the points $3\sqrt{2}$ and $p\sqrt{2}i$. Let $z_1 = 3\sqrt{2}$ and $z_2 = p\sqrt{2}i$. In the complex plane, $z_1$ corresponds to the point $(3\sqrt{2}, 0)$ and $z_2$ corresponds to the point $(0, p\sqrt{2})$.

Step 2: Apply the triangle inequality to find the minimum value.

The minimum value of $|z - 3\sqrt{2}| + |z - p\sqrt{2}i|$ is the distance between the points $3\sqrt{2}$ and $p\sqrt{2}i$. This minimum value is given as $5\sqrt{2}$. Thus, we have $|3\sqrt{2} - p\sqrt{2}i| = 5\sqrt{2}$

Step 3: Calculate the distance between the two points using the distance formula.

The distance between the points $(3\sqrt{2}, 0)$ and $(0, p\sqrt{2})$ is $\sqrt{(0 - 3\sqrt{2})^2 + (p\sqrt{2} - 0)^2} = \sqrt{(-3\sqrt{2})^2 + (p\sqrt{2})^2} = \sqrt{18 + 2p^2}$

Step 4: Set up the equation and solve for p.

We are given that the minimum value is $5\sqrt{2}$. Therefore, $\sqrt{18 + 2p^2} = 5\sqrt{2}$ Squaring both sides, we get $18 + 2p^2 = (5\sqrt{2})^2 = 25 \cdot 2 = 50$ $2p^2 = 50 - 18 = 32$ $p^2 = \frac{32}{2} = 16$ $p = \pm \sqrt{16} = \pm 4$

Step 5: Determine a value for p from the options.

Since the question asks for "a value of $p$", we can choose either $4$ or $-4$. The options provided include 4.

Common Mistakes & Tips

• Forgetting the $\pm$ sign: When taking the square root to solve for $p$, remember to consider both positive and negative roots.
• Misinterpreting Modulus: The expression $|z - a|$ represents the distance between the point $z$ and the point $a$ in the complex plane.
• Squaring Square Roots: Be careful when squaring expressions involving square roots. Ensure that both the numerical and radical parts are squared correctly.

Summary

The problem involves finding a value of $p$ given the minimum distance between a complex number $z$ and two fixed points in the complex plane. By recognizing the geometric interpretation of the modulus and applying the distance formula, we set up an equation and solve for $p$. We found that $p = \pm 4$, and the given options include $p=4$.

Final Answer

The final answer is \boxed{4}, which corresponds to option (C).