If z is complex number such that `|z|ge2` then the minimum value of `|z+1/(2)|`

Options

A. is equal to `5/(2)`
B. lies in the interval [1, 2]
C. is strictly greater than `3/(2)`
D. is strictly greater than `3/(2)` but less than `5/(2)`

Answer

To find the minimum value of \( |z + \frac{1}{2}| \) given that \( |z| \geq 2 \), we can follow these steps:

### Step 1: Understand the given condition
We know that \( |z| \geq 2 \). This means that the complex number \( z \) lies outside or on the boundary of a circle centered at the origin (0,0) with a radius of 2.

### Step 2: Rewrite the expression
We need to minimize \( |z + \frac{1}{2}| \). This can be interpreted as finding the distance from the point \( -\frac{1}{2} \) (which is the point corresponding to \( \frac{1}{2} \) on the real axis) to the points \( z \) that are at least 2 units away from the origin.

### Step 3: Visualize the situation
The point \( -\frac{1}{2} \) is located at \( (-0.5, 0) \) on the complex plane. The circle defined by \( |z| = 2 \) has its center at the origin and a radius of 2.

### Step 4: Determine the closest point on the circle to \( -\frac{1}{2} \)
To find the minimum distance from \( -\frac{1}{2} \) to the circle \( |z| = 2 \), we need to find the point on the circle that is closest to \( -\frac{1}{2} \).

1. The distance from the origin to \( -\frac{1}{2} \) is \( |-\frac{1}{2}| = \frac{1}{2} \).
2. The distance from the origin to the circle is 2.
3. The closest point on the circle to \( -\frac{1}{2} \) will be along the line connecting the origin to \( -\frac{1}{2} \).

### Step 5: Calculate the distance
The distance from \( -\frac{1}{2} \) to the circle can be calculated as follows:
- The distance from the origin to \( -\frac{1}{2} \) is \( \frac{1}{2} \).
- The radius of the circle is 2.
- Therefore, the distance from \( -\frac{1}{2} \) to the circle is \( 2 - \frac{1}{2} = \frac{3}{2} \).

### Step 6: Find the minimum value
Thus, the minimum value of \( |z + \frac{1}{2}| \) when \( |z| \geq 2 \) is:
\[ \text{Minimum value} = 2 - \frac{1}{2} = \frac{3}{2} \]

### Final Answer
The minimum value of \( |z + \frac{1}{2}| \) is \( \frac{3}{2} \).