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

Answer: (A)

12

Answer 2

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 $|z - (6 + 8i)|$:

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