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: Analyze the expression
Now, we can analyze the expression \( 8|Z - 1| + 2|Z - 2| \). Here, \( |Z - 1| \) represents the distance from the point \( Z \) to the point \( 1 \) in the complex plane, and \( |Z - 2| \) represents the distance from the point \( Z \) to the point \( 2 \).

### Step 3: Use the triangle inequality
To minimize the expression, we can use the triangle inequality. The minimum value occurs when the points \( Z \), \( 1 \), and \( 2 \) are collinear. This means that \( Z \) should lie on the line segment connecting \( 1 \) and \( 2 \).

### Step 4: Evaluate the expression at critical points
We will evaluate the expression at the endpoints of the segment, which are \( Z = 1 \) and \( Z = 2 \).

1. **At \( Z = 1 \)**:
\[ |8(1) - 8| + |2(1) - 4| = |0| + |-2| = 0 + 2 = 2 \]

2. **At \( Z = 2 \)**:
\[ |8(2) - 8| + |2(2) - 4| = |16 - 8| + |4 - 4| = |8| + |0| = 8 + 0 = 8 \]

### Step 5: Determine the minimum value
From the evaluations, we see that:
- At \( Z = 1 \), the value is \( 2 \).
- At \( Z = 2 \), the value is \( 8 \).

Thus, the minimum value of the expression occurs at \( Z = 1 \).

### Conclusion
The minimum value of \( |8Z - 8| + |2Z - 4| \) exists when \( Z = 1 \).