The sides of a square are `x=2,x=3,y=1andy=2`. Find the equation of the circle drawn on the diagonals of the square as its diameter.

Answer

To find the equation of the circle drawn on the diagonals of the square as its diameter, we will follow these steps:

### Step 1: Identify the vertices of the square
Given the lines:
- \( x = 2 \)
- \( x = 3 \)
- \( y = 1 \)
- \( y = 2 \)

The vertices of the square can be determined by the intersections of these lines:
- Bottom left vertex \( A(2, 1) \)
- Bottom right vertex \( B(3, 1) \)
- Top right vertex \( C(3, 2) \)
- Top left vertex \( D(2, 2) \)

### Step 2: Identify the endpoints of the diagonal
We can choose diagonal \( AC \) or \( BD \). Let's choose diagonal \( AC \):
- \( A(2, 1) \)
- \( C(3, 2) \)

### Step 3: Find the midpoint of the diagonal
The midpoint \( M \) of diagonal \( AC \) is calculated as follows:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 3}{2}, \frac{1 + 2}{2} \right) = \left( \frac{5}{2}, \frac{3}{2} \right) \]

### Step 4: Calculate the length of the diagonal
The length \( d \) of diagonal \( AC \) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 5: Determine the radius of the circle
The radius \( r \) of the circle is half the length of the diagonal:
\[ r = \frac{d}{2} = \frac{\sqrt{2}}{2} \]

### Step 6: Write the equation of the circle
The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = \frac{5}{2} \), \( k = \frac{3}{2} \), and \( r = \frac{\sqrt{2}}{2} \):
\[ (x - \frac{5}{2})^2 + (y - \frac{3}{2})^2 = \left(\frac{\sqrt{2}}{2}\right)^2 \]
\[ (x - \frac{5}{2})^2 + (y - \frac{3}{2})^2 = \frac{2}{4} = \frac{1}{2} \]

### Final Equation
Thus, the equation of the circle is:
\[ (x - \frac{5}{2})^2 + (y - \frac{3}{2})^2 = \frac{1}{2} \]