Question

If \( z = 1 + i \), then the maximum value of \( |z + 12 + 9i| \) is

• A. 225
• B. 265
• C. \( \sqrt{269} \)
• D. 200
• E. \( \sqrt{265} \)

Hint:

The modulus of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \).

Answer 1

The Correct Option is C

Step 1: Add \( 12 + 9i \) to \( z = 1 + i \): \[ z + 12 + 9i = 1 + i + 12 + 9i = 13 + 10i \] Step 2: Now, calculate the modulus: \[ |13 + 10i| = \sqrt{13^2 + 10^2} = \sqrt{169 + 100} = \sqrt{269} \] Step 3: Thus, the maximum value is \( \sqrt{269} \).