Question

Let \( z \) be a complex number such that \( |z| = 1 \). If \[ \frac{2 + kz}{k + z} = kz,\ k \in \mathbb{R}, \] then the maximum distance of \( k + ik^2 \) from the circle \( |z - (1 + 2i)| = 1 \) is:

• A. \( \sqrt{5} + 1 \)
• B. 2
• C. 3
• D. \( \sqrt{5} + \sqrt{1} \)

Hint:

Use geometry and coordinate form of complex numbers to compute distances.

Answer 1

The Correct Option is A

Given: \[ \frac{2 + kz}{k + z} = kz \Rightarrow (2 + kz) = kz(k + z) \Rightarrow 2 + kz = k^2z + kz^2 \] Solving and using \( |z| = 1 \Rightarrow zz^* = 1 \) gives \( k = 2 \) Now, find distance between \( (k, k^2) = (2, 4) \) and center \( (1, 2) \): \[ \text{Distance} = \sqrt{(2 - 1)^2 + (4 - 2)^2} = \sqrt{1 + 4} = \sqrt{5} \] Max distance from boundary = \( \sqrt{5} + 1 \)