To find the equation of the circle drawn on the diagonals of the rectangle as its diameter, we will follow these steps for both parts of the question.
### Part (i): Rectangle with sides x = 6, x = -3, y = 3, and y = -1
1. **Identify the vertices of the rectangle**:
- The vertices can be determined from the given lines:
- A = (-3, -1)
- B = (-3, 3)
- C = (6, 3)
- D = (6, -1)
2. **Find the coordinates of the endpoints of the diagonal**:
- The endpoints of the diagonal are A and C:
- A = (-3, -1)
- C = (6, 3)
3. **Calculate the midpoint of the diagonal (center of the circle)**:
- The midpoint \(O\) can be calculated using the midpoint formula:
\[
O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 6}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{3}{2}, 1 \right)
\]
4. **Calculate the radius of the circle**:
- The radius \(r\) is half the length of the diagonal. We can find the length of the diagonal using the distance formula:
\[
r = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \frac{1}{2} \sqrt{(6 - (-3))^2 + (3 - (-1))^2}
\]
\[
= \frac{1}{2} \sqrt{(6 + 3)^2 + (3 + 1)^2} = \frac{1}{2} \sqrt{9^2 + 4^2} = \frac{1}{2} \sqrt{81 + 16} = \frac{1}{2} \sqrt{97}
\]
5. **Write the equation of the circle**:
- The standard equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Substituting \(h = \frac{3}{2}\), \(k = 1\), and \(r = \frac{\sqrt{97}}{2}\):
\[
\left(x - \frac{3}{2}\right)^2 + (y - 1)^2 = \left(\frac{\sqrt{97}}{2}\right)^2
\]
\[
\left(x - \frac{3}{2}\right)^2 + (y - 1)^2 = \frac{97}{4}
\]
### Part (ii): Rectangle with sides x = 5, x = 8, y = 4, and y = 7
1. **Identify the vertices of the rectangle**:
- The vertices can be determined from the given lines:
- A = (5, 4)
- B = (5, 7)
- C = (8, 7)
- D = (8, 4)
2. **Find the coordinates of the endpoints of the diagonal**:
- The endpoints of the diagonal are A and C:
- A = (5, 4)
- C = (8, 7)
3. **Calculate the midpoint of the diagonal (center of the circle)**:
- The midpoint \(O\) can be calculated using the midpoint formula:
\[
O = \left( \frac{5 + 8}{2}, \frac{4 + 7}{2} \right) = \left( \frac{13}{2}, \frac{11}{2} \right)
\]
4. **Calculate the radius of the circle**:
- The radius \(r\) is half the length of the diagonal. We can find the length of the diagonal using the distance formula:
\[
r = \frac{1}{2} \sqrt{(8 - 5)^2 + (7 - 4)^2} = \frac{1}{2} \sqrt{3^2 + 3^2} = \frac{1}{2} \sqrt{9 + 9} = \frac{1}{2} \sqrt{18} = \frac{3\sqrt{2}}{2}
\]
5. **Write the equation of the circle**:
- The standard equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
- Substituting \(h = \frac{13}{2}\), \(k = \frac{11}{2}\), and \(r = \frac{3\sqrt{2}}{2}\):
\[
\left(x - \frac{13}{2}\right)^2 + \left(y - \frac{11}{2}\right)^2 = \left(\frac{3\sqrt{2}}{2}\right)^2
\]
\[
\left(x - \frac{13}{2}\right)^2 + \left(y - \frac{11}{2}\right)^2 = \frac{9 \cdot 2}{4} = \frac{18}{4} = \frac{9}{2}
\]
### Summary of Solutions:
- For part (i), the equation of the circle is:
\[
\left(x - \frac{3}{2}\right)^2 + (y - 1)^2 = \frac{97}{4}
\]
- For part (ii), the equation of the circle is:
\[
\left(x - \frac{13}{2}\right)^2 + \left(y - \frac{11}{2}\right)^2 = \frac{9}{2}
\]
