If `z epsilon C`, the minimum value of `|z| + |z-i|` is attained at

Options

A. exactly one point
B. exactly two points
C. infinite number of points
D. None of these

Answer

To find the minimum value of \( |z| + |z - i| \) for \( z \in \mathbb{C} \), we can follow these steps:

### Step 1: Define the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Express the moduli
We can express the moduli as follows:
- \( |z| = |x + iy| = \sqrt{x^2 + y^2} \)
- \( |z - i| = |x + iy - i| = |x + i(y - 1)| = \sqrt{x^2 + (y - 1)^2} \)

### Step 3: Set up the expression to minimize
We need to minimize the expression:
\[ |z| + |z - i| = \sqrt{x^2 + y^2} + \sqrt{x^2 + (y - 1)^2} \]

### Step 4: Use the triangle inequality
Using the triangle inequality, we know that:
\[ |z| + |z - i| \geq |z - 0 + (0 - i)| = |z - i| \]
The equality holds when \( z \) lies on the line segment connecting \( 0 \) and \( i \).

### Step 5: Find the minimum value
The minimum value of \( |z| + |z - i| \) occurs when \( z \) is on the line segment between \( 0 \) and \( i \). The distance between these two points is:
\[ |0 - i| = 1 \]
Thus, the minimum value of \( |z| + |z - i| \) is \( 1 \).

### Step 6: Determine the points where the minimum occurs
The minimum occurs when \( z \) is any point on the line segment joining \( 0 \) and \( i \). This can be expressed as:
\[ z = 0 + it \quad \text{for } 0 \leq t \leq 1 \]
This means \( z \) can take any value of the form \( iy \) where \( y \) is between \( 0 \) and \( 1 \).

### Conclusion
The minimum value of \( |z| + |z - i| \) is \( 1 \), and it is attained at an infinite number of points along the line segment from \( 0 \) to \( i \).

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