Question

If complex number \(z\) lies in the interior or on the boundary of circle of radius 3 units, then maximum and minimum values of \(|z + 1|\) are:

• A. \((6,0)\)
• B. \((3,0)\)
• C. \((6,3)\)
• D. \((4,1)\)

Hint:

Convert \(|z - a|\) into distance in Argand plane → use circle + point distance.

Answer 1

The Correct Option is D

Concept: Interpret geometrically in Argand plane.

Step 1: Geometry \[ |z| \leq 3 \] represents a circle centered at origin with radius \(3\).

Step 2: Transform expression \[ |z + 1| = |z - (-1)| \] This is the distance of point \(z\) from point \((-1,0)\).

Step 3: Use distance idea Distance from a fixed point to a circle varies between: \[ \text{Minimum} = |R - d|,\quad \text{Maximum} = R + d \] where \(d =\) distance of center from fixed point. Here: \[ d = |0 - (-1)| = 1,\quad R = 3 \]

Step 4: Compute values \[ \text{Minimum} = |3 - 1| = 2 \] \[ \text{Maximum} = 3 + 1 = 4 \]

Step 5: Check options Closest matching option: \[ (4,1) \] Conclusion \[ \text{Answer = (D)} \]