Question

For $z \in C$ if the minimum value of $(| z -3 \sqrt{2}|+| z - p \sqrt{2} i |)$ is $5 \sqrt{2}$, then a value of $p$ is ______

• A. 3
• B. $\frac{7}{2}$
• C. 4
• D. $\frac{9}{2}$

Answer 1

The Correct Option is C

To solve the problem of finding the value of \( p \) for which the minimum value of the expression \( |z - 3\sqrt{2}| + |z - p\sqrt{2}i| \) is \( 5\sqrt{2} \), we can utilize the geometrical interpretation of the expression involving complex numbers.

The given expression represents the sum of distances from a point \( z \) in the complex plane to two fixed points: the point \( A(3\sqrt{2}, 0) \) and the point \( B(0, p\sqrt{2}) \). The minimum value condition for the sum of these distances is analogous to finding the minimum for the sum of distances from a point on a straight line connecting two points, which occurs when the point is on the line segment itself, ensuring that \( |z - A| + |z - B| = AB \).

Using the distance formula, the distance between points \( A(3\sqrt{2}, 0) \) and \( B(0, p\sqrt{2}) \) is:

\(AB = \sqrt{(3\sqrt{2} - 0)^2 + (0 - p\sqrt{2})^2}\)

Thus,

\(AB = \sqrt{(3\sqrt{2})^2 + (p\sqrt{2})^2} = \sqrt{18 + 2p^2}\)

For the minimum value \( 5\sqrt{2} \) to occur, we set:

\(\sqrt{18 + 2p^2} = 5\sqrt{2}\)

Squaring both sides results in:

\(18 + 2p^2 = 50\)

Solving for \( p^2 \), we have:

\(2p^2 = 32 \Rightarrow p^2 = 16 \Rightarrow p = \sqrt{16} = 4\)

Thus, the value of \( p \) is 4. Let's verify whether other options could be correct. The remaining options include fractions and integers other than 4, whose application would not satisfy the condition derived above. Thus, among the options provided, \( p = 4 \) is the correct answer.

Therefore, the correct option is 4.