The chords through (2,1) to the circle `x^2-2x+y^2-2y+1=0` are bisected at the point `(alpha,1/2)` then value of `(2alpha)/3` is ___

Answer

To solve the problem, we need to find the value of \( \frac{2\alpha}{3} \) where the chords through the point \( (2, 1) \) to the circle given by the equation \( x^2 - 2x + y^2 - 2y + 1 = 0 \) are bisected at the point \( (\alpha, \frac{1}{2}) \).

### Step-by-Step Solution:

1. **Rewrite the Circle Equation**:
The given circle equation is:
\[ x^2 - 2x + y^2 - 2y + 1 = 0 \]
We can complete the square for both \( x \) and \( y \):
\[ (x-1)^2 + (y-1)^2 = 1 \]
This represents a circle with center \( (1, 1) \) and radius \( 1 \).

2. **Identify the Midpoint of the Chord**:
The midpoint of the chord is given as \( (\alpha, \frac{1}{2}) \).

3. **Use the Midpoint Formula**:
The equation of the chord can be derived using the midpoint formula. If \( (x_1, y_1) \) and \( (x_2, y_2) \) are the endpoints of the chord, then:
\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \alpha, \frac{1}{2} \right) \]
Thus, we have:
\[ x_1 + x_2 = 2\alpha \quad \text{and} \quad y_1 + y_2 = 1 \]

4. **Substitute the Point (2, 1)**:
Since the chord passes through the point \( (2, 1) \), we can let one endpoint be \( (2, 1) \) and the other endpoint be \( (x, y) \). Therefore:
\[ 2 + x = 2\alpha \quad \text{and} \quad 1 + y = 1 \]
From \( 1 + y = 1 \), we get \( y = 0 \).

5. **Solve for \( x \)**:
From \( 2 + x = 2\alpha \):
\[ x = 2\alpha - 2 \]

6. **Equation of the Chord**:
The endpoints of the chord are \( (2, 1) \) and \( (2\alpha - 2, 0) \). The slope \( m \) of the chord is:
\[ m = \frac{0 - 1}{(2\alpha - 2) - 2} = \frac{-1}{2\alpha - 4} \]
The equation of the line (chord) can be written using point-slope form:
\[ y - 1 = m(x - 2) \]

7. **Substitute into Circle Equation**:
Substitute \( y = \frac{-1}{2\alpha - 4}(x - 2) + 1 \) into the circle equation:
\[ (x - 1)^2 + \left(\frac{-1}{2\alpha - 4}(x - 2) + 1 - 1\right)^2 = 1 \]

8. **Simplify and Solve for \( \alpha \)**:
After substituting and simplifying, we will end up with a quadratic equation in terms of \( \alpha \). Solving this quadratic will give us the value of \( \alpha \).

9. **Calculate \( \frac{2\alpha}{3} \)**:
Once we find \( \alpha \), we can compute \( \frac{2\alpha}{3} \).

### Final Calculation:
After solving the quadratic equation, we find \( \alpha = \frac{3}{2} \). Therefore:
\[ \frac{2\alpha}{3} = \frac{2 \times \frac{3}{2}}{3} = 1 \]

### Answer:
The value of \( \frac{2\alpha}{3} \) is \( \boxed{1} \).