If "z" is a complex number such that `|z| ge 1` ,then the minimum value of `|z+(1)/(2)(3+4i)|` is :

Options

A. `3/2`
B. 3
C. `5/2`
D. 2

Answer

To find the minimum value of \( |z + \frac{1}{2}(3 + 4i)| \) given that \( |z| \geq 1 \), we can follow these steps:

### Step 1: Rewrite the expression
We need to analyze the expression \( |z + \frac{1}{2}(3 + 4i)| \). First, we can simplify \( \frac{1}{2}(3 + 4i) \):
\[ \frac{1}{2}(3 + 4i) = \frac{3}{2} + 2i \]
Thus, we need to find the minimum of:
\[ |z + \left(\frac{3}{2} + 2i\right)| \]

### Step 2: Interpret the problem geometrically
The condition \( |z| \geq 1 \) defines a region outside (and including) a circle of radius 1 centered at the origin in the complex plane. The point \( \frac{3}{2} + 2i \) can be represented as a point in the complex plane.

### Step 3: Calculate the distance from the center of the circle to the point
Next, we need to find the distance from the point \( \frac{3}{2} + 2i \) to the origin \( 0 + 0i \):
\[ \text{Distance} = \sqrt{\left(\frac{3}{2}\right)^2 + (2)^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{9}{4} + \frac{16}{4}} = \sqrt{\frac{25}{4}} = \frac{5}{2} \]

### Step 4: Determine the minimum distance from the circle
The minimum distance from the circle (which has a radius of 1) to the point \( \frac{3}{2} + 2i \) is given by subtracting the radius of the circle from the distance calculated:
\[ \text{Minimum Distance} = \frac{5}{2} - 1 = \frac{3}{2} \]

### Step 5: Conclusion
Thus, the minimum value of \( |z + \frac{1}{2}(3 + 4i)| \) is:
\[ \frac{3}{2} \]