the minimum value of `|8Z-8|+|2Z-4|` exists, when Z is equal to (where, Z is a complex number)

Options

A. 2
B. 1.5
C. 0
D. 1

Answer

To find the minimum value of the expression \( |8Z - 8| + |2Z - 4| \) where \( Z \) is a complex number, we can follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression:
\[ |8Z - 8| + |2Z - 4| = |8(Z - 1)| + |2(Z - 2)| \]
This simplifies to:
\[ 8|Z - 1| + 2|Z - 2| \]

### Step 2: Factor Out Constants
We can factor out the constants from the absolute values:
\[ = 8|Z - 1| + 2|Z - 2| \]

### Step 3: Analyze the Absolute Values
To minimize the expression \( 8|Z - 1| + 2|Z - 2| \), we need to consider the points \( Z = 1 \) and \( Z = 2 \). The expression is composed of two terms, and we can analyze their contributions to the total.

### Step 4: Evaluate at Key Points
1. **Evaluate at \( Z = 1 \)**:
\[ 8|1 - 1| + 2|1 - 2| = 8 \cdot 0 + 2 \cdot 1 = 2 \]

2. **Evaluate at \( Z = 2 \)**:
\[ 8|2 - 1| + 2|2 - 2| = 8 \cdot 1 + 2 \cdot 0 = 8 \]

### Step 5: Compare Values
From the evaluations:
- At \( Z = 1 \), the value is \( 2 \).
- At \( Z = 2 \), the value is \( 8 \).

### Step 6: Conclusion
The minimum value occurs at \( Z = 1 \), where the expression evaluates to \( 2 \).

Thus, the minimum value of \( |8Z - 8| + |2Z - 4| \) exists when \( Z = 1 \).

### Final Answer
\[ Z = 1 \]