If centre of the circle lies on `x+y=2` and it touches the line `x=3` and `y=2`. Find the dismeter of the circle.

Answer

To find the diameter of the circle given that its center lies on the line \(x + y = 2\) and it touches the lines \(x = 3\) and \(y = 2\), we can follow these steps:

### Step 1: Understand the Position of the Circle
The center of the circle, denoted as \((h, k)\), lies on the line \(x + y = 2\). This means that:
\[ h + k = 2 \]

### Step 2: Determine the Radius
Since the circle touches the line \(x = 3\), the distance from the center \((h, k)\) to the line \(x = 3\) is equal to the radius \(r\). The distance from a point \((h, k)\) to the line \(x = 3\) is given by:
\[ \text{Distance} = |h - 3| \]
Thus, we have:
\[ r = |h - 3| \]

Similarly, since the circle touches the line \(y = 2\), the distance from the center \((h, k)\) to the line \(y = 2\) is also equal to the radius \(r\). The distance from a point \((h, k)\) to the line \(y = 2\) is given by:
\[ \text{Distance} = |k - 2| \]
Thus, we have:
\[ r = |k - 2| \]

### Step 3: Set Up the Equations
From the two equations we have:
1. \(r = |h - 3|\)
2. \(r = |k - 2|\)

Since both expressions equal \(r\), we can set them equal to each other:
\[ |h - 3| = |k - 2| \]

### Step 4: Substitute \(k\) in Terms of \(h\)
From the equation \(h + k = 2\), we can express \(k\) in terms of \(h\):
\[ k = 2 - h \]

### Step 5: Substitute \(k\) into the Distance Equation
Substituting \(k\) into the distance equation:
\[ |h - 3| = |(2 - h) - 2| \]
This simplifies to:
\[ |h - 3| = | - h | \]
Thus, we have:
\[ |h - 3| = |h| \]

### Step 6: Solve the Absolute Value Equations
This gives us two cases to consider:
1. \(h - 3 = h\) (which is not possible)
2. \(h - 3 = -h\)

From the second case:
\[ h - 3 = -h \implies 2h = 3 \implies h = \frac{3}{2} \]

### Step 7: Find \(k\)
Substituting \(h = \frac{3}{2}\) back into the equation for \(k\):
\[ k = 2 - h = 2 - \frac{3}{2} = \frac{1}{2} \]

### Step 8: Find the Radius
Now we can find the radius \(r\):
\[ r = |h - 3| = \left|\frac{3}{2} - 3\right| = \left|-\frac{3}{2}\right| = \frac{3}{2} \]

### Step 9: Find the Diameter
The diameter \(D\) of the circle is twice the radius:
\[ D = 2r = 2 \times \frac{3}{2} = 3 \]

Thus, the diameter of the circle is:
\[ \boxed{3} \]