Question

Let \( z \) be a complex number such that the real part of \[ \frac{z - 2i}{z + 2i} \] is zero. Then, the maximum value of \( |z - (6 + 8i)| \) is equal to:

• A. 12
• B. \(\infty\)
• C. 10
• D. 8

Answer 1

The Correct Option is A

Given the expression:

\[ \frac{z - 2i}{z + 2i} + \frac{\overline{z} + 2i}{\overline{z} - 2i} = 0, \]

we proceed by simplifying each term. Expanding and multiplying, we obtain:

\[ z\overline{z} - 2i\overline{z} - 2iz + 4(-1) + \overline{z}z + 2zi + 2z\overline{i} + 4(-1) = 0. \]

Combining terms, we get:

\[ 2|z|^2 = 8 \implies |z| = 2. \]

Now, we find the maximum value of \( |z - (6 + 8i)| \):

\[ |z - (6 + 8i)|_{\text{maximum}} = 10 + 2 = 12. \]