The sides of a rectangle are given by the equations x=-2, x = 4, y=-2 andy=5. Find the equation of the circle drawn on the diagonal of this rectangle as its diameter.

Answer

To find the equation of the circle drawn on the diagonal of the rectangle as its diameter, we can follow these steps:

### Step 1: Identify the vertices of the rectangle
The sides of the rectangle are given by the equations:
- \( x = -2 \)
- \( x = 4 \)
- \( y = -2 \)
- \( y = 5 \)

The vertices of the rectangle can be determined by the intersection of these lines:
- Bottom-left vertex: \( (-2, -2) \)
- Bottom-right vertex: \( (4, -2) \)
- Top-left vertex: \( (-2, 5) \)
- Top-right vertex: \( (4, 5) \)

### Step 2: Identify the endpoints of the diagonal
The diagonal of the rectangle can be taken as the line segment connecting the bottom-left vertex \( A(-2, -2) \) and the top-right vertex \( B(4, 5) \).

### Step 3: Find the midpoint of the diagonal
The midpoint \( M \) of the diagonal \( AB \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ M = \left( \frac{-2 + 4}{2}, \frac{-2 + 5}{2} \right) = \left( \frac{2}{2}, \frac{3}{2} \right) = (1, 1.5) \]

### Step 4: Calculate the radius of the circle
The radius \( r \) of the circle is half the length of the diagonal \( AB \). The length of the diagonal can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{(4 - (-2))^2 + (5 - (-2))^2} = \sqrt{(4 + 2)^2 + (5 + 2)^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85} \]
Thus, the radius \( r \) is:
\[ r = \frac{AB}{2} = \frac{\sqrt{85}}{2} \]

### Step 5: Write the equation of the circle
The standard equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 1 \), \( k = 1.5 \), and \( r = \frac{\sqrt{85}}{2} \):
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \left(\frac{\sqrt{85}}{2}\right)^2 \]
Calculating \( r^2 \):
\[ r^2 = \left(\frac{\sqrt{85}}{2}\right)^2 = \frac{85}{4} \]
Thus, the equation of the circle is:
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{85}{4} \]

### Final Answer
The equation of the circle drawn on the diagonal of the rectangle as its diameter is:
\[ (x - 1)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{85}{4} \]