If one end of the diameter is (1, 1) and the other end lies on the line `x+y=3` , then find the locus of the center of the circle.

Explanation

To find the locus of the center of the circle given that one end of the diameter is at point A(1, 1) and the other end lies on the line x + y = 3, we can follow these steps:

### Step 1: Define the Points
Let the coordinates of the center of the circle be \( C(h, k) \). Since A(1, 1) is one end of the diameter, we need to find the coordinates of the other end, which we will denote as B.

### Step 2: Find the Coordinates of Point B
The coordinates of point B can be expressed in terms of the center C(h, k). The relationship between the center and the endpoints of the diameter is given by:
\[ C(h, k) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) are the coordinates of point A and \( (x_2, y_2) \) are the coordinates of point B.

Substituting point A(1, 1):
\[ h = \frac{1 + x_2}{2} \quad \text{and} \quad k = \frac{1 + y_2}{2} \]
From these equations, we can express \( x_2 \) and \( y_2 \) in terms of \( h \) and \( k \):
\[ x_2 = 2h - 1 \quad \text{and} \quad y_2 = 2k - 1 \]

### Step 3: Use the Condition of Point B on the Line
Since point B lies on the line \( x + y = 3 \), we can substitute the coordinates of B into this equation:
\[ (2h - 1) + (2k - 1) = 3 \]

### Step 4: Simplify the Equation
Now, simplify the equation:
\[ 2h - 1 + 2k - 1 = 3 \]
\[ 2h + 2k - 2 = 3 \]
\[ 2h + 2k = 5 \]

### Step 5: Write the Equation of the Locus
To express the locus, we can replace \( h \) with \( x \) and \( k \) with \( y \):
\[ 2x + 2y = 5 \]
Dividing the entire equation by 2 gives:
\[ x + y = \frac{5}{2} \]

### Final Answer
Thus, the locus of the center of the circle is:
\[ x + y = \frac{5}{2} \]