If z is a complex number such that `abs(z)le1` then minimum value of `abs(z+1/2(3+4i))` is equal to

Options

A. 3
B. 2
C. `3/2`
D. `1/2`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ |z + \frac{1}{2}(3 + 4i)| \]
This can be rewritten as:
\[ |z + \frac{3}{2} + 2i| \]
Let \( w = z + \frac{3}{2} + 2i \).

### Step 2: Identify the center of the circle
The term \( \frac{3}{2} + 2i \) represents a point in the complex plane. The center of the circle defined by \( |z| \leq 1 \) is at the origin \( (0, 0) \).

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

### Step 4: Determine the minimum distance
Since the radius of the circle defined by \( |z| \leq 1 \) is 1, the minimum distance from the point \( \left(\frac{3}{2}, 2\right) \) to the circle is:
\[ \text{Minimum Distance} = \text{Distance from center to point} - \text{Radius} = \frac{5}{2} - 1 = \frac{3}{2} \]

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