Question

If \( \alpha \) is the modulus of \( z_1 = 4 + 3i \), then a point that does not lie in the region represented by } \[ |z - z_1| \leq \alpha \] is

• A. \( z_1 - 2i \)
• B. \( z_1 \)
• C. \( 2z_1 - 7i \)
• D. \( 3z_1 - (10 + 8i) \)

Hint:

Inequality \( |z - z_0| \leq r \) defines a closed disk of radius \( r \) centered at \( z_0 \). Points farther than \( r \) from the center lie outside.

Answer 1

The Correct Option is B

Step 1: Compute \( \alpha = |z_1| \)
\[ z_1 = 4 + 3i \Rightarrow \alpha = |z_1| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \] Step 2: Identify region
The inequality \( |z - z_1| \leq 5 \) describes a closed disk of radius 5 centered at \( z_1 \). Step 3: Test all options
- Option (2): \( z = z_1 \Rightarrow |z - z_1| = 0 \leq 5 \) ⇒ lies inside ✅ - Others exceed the radius or lie outside ⇒ ❌