The minimum value of k for which `|z-i|^2 + |z-1|^2 =k` will represent a circle is ____________.

Answer

To find the minimum value of \( k \) for which the equation \( |z - i|^2 + |z - 1|^2 = k \) represents a circle, we will follow these steps:

### Step 1: Express \( z \) in terms of \( x \) and \( y \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the equation using the definition of modulus
The equation becomes:
\[ |z - i|^2 + |z - 1|^2 = |(x + iy) - i|^2 + |(x + iy) - 1|^2 \]
This simplifies to:
\[ |(x + i(y - 1))|^2 + |((x - 1) + iy)|^2 \]

### Step 3: Calculate the moduli
Calculating the moduli gives:
\[ |(x + i(y - 1))|^2 = x^2 + (y - 1)^2 \]
\[ |((x - 1) + iy)|^2 = (x - 1)^2 + y^2 \]

### Step 4: Combine the equations
Now, substituting these into the equation gives:
\[ x^2 + (y - 1)^2 + (x - 1)^2 + y^2 = k \]

### Step 5: Expand the terms
Expanding the terms:
\[ x^2 + (y^2 - 2y + 1) + (x^2 - 2x + 1) + y^2 = k \]
This simplifies to:
\[ 2x^2 + 2y^2 - 2x - 2y + 2 = k \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 2x^2 + 2y^2 - 2x - 2y + (2 - k) = 0 \]
Dividing through by 2:
\[ x^2 + y^2 - x - y + \frac{2 - k}{2} = 0 \]

### Step 7: Identify the circle's parameters
This is in the form of a circle equation:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g = -\frac{1}{2} \), \( f = -\frac{1}{2} \), and \( c = \frac{2 - k}{2} \).

### Step 8: Find the condition for a circle
For this to represent a circle, the condition is:
\[ g^2 + f^2 - c \geq 0 \]
Calculating \( g^2 + f^2 \):
\[ g^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}, \quad f^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]
Thus:
\[ g^2 + f^2 = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \]
Now substituting for \( c \):
\[ \frac{1}{2} - \frac{2 - k}{2} \geq 0 \]
Simplifying gives:
\[ \frac{1}{2} - 1 + \frac{k}{2} \geq 0 \]
This leads to:
\[ \frac{k}{2} \geq \frac{1}{2} \]
Thus:
\[ k \geq 1 \]

### Conclusion
The minimum value of \( k \) for which the equation represents a circle is:
\[ \boxed{1} \]