A circle touch the line `x=3` and `y=2` and a line `x+y=2` passes through its centre the diameter of a circle is

Options

A. 4
B. 3
C. 2
D. 5

Answer

To solve the problem, we need to find the diameter of a circle that touches the lines \(x = 3\) and \(y = 2\), and whose center lies on the line \(x + y = 2\).

### Step-by-Step Solution:

1. **Understanding the Circle's Position**:
- The circle touches the line \(x = 3\). This means that the distance from the center of the circle to the line \(x = 3\) is equal to the radius \(r\) of the circle.
- The circle also touches the line \(y = 2\). This means that the distance from the center of the circle to the line \(y = 2\) is also equal to the radius \(r\).

2. **Setting Up the Center Coordinates**:
- Let the center of the circle be at the point \((h, k)\).
- Since the circle touches the line \(x = 3\), the distance from the center to this line is given by:
\[ |h - 3| = r \]
- Since the circle touches the line \(y = 2\), the distance from the center to this line is given by:
\[ |k - 2| = r \]

3. **Using the Line Equation**:
- The center of the circle also lies on the line \(x + y = 2\). Therefore, we have:
\[ h + k = 2 \]

4. **Expressing the Center Coordinates**:
- From the first condition, we can express \(h\) in terms of \(r\):
\[ h = 3 - r \quad \text{(since the circle touches the line from the left)} \]
- From the second condition, we can express \(k\) in terms of \(r\):
\[ k = 2 - r \quad \text{(since the circle touches the line from below)} \]

5. **Substituting into the Line Equation**:
- Now, substituting \(h\) and \(k\) into the line equation \(h + k = 2\):
\[ (3 - r) + (2 - r) = 2 \]
- Simplifying this gives:
\[ 5 - 2r = 2 \]
- Rearranging gives:
\[ 2r = 3 \quad \Rightarrow \quad r = \frac{3}{2} \]

6. **Finding the Diameter**:
- The diameter \(D\) of the circle is twice the radius:
\[ D = 2r = 2 \times \frac{3}{2} = 3 \]

### Final Answer:
The diameter of the circle is \(3\).