To find the equation of the circle passing through three given points, we can use the general equation of a circle:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
where \(g\), \(f\), and \(c\) are constants. The center of the circle is given by \((-g, -f)\) and the radius can be calculated using the formula:
\[
\text{Radius} = \sqrt{g^2 + f^2 - c}
\]
Now, let's solve each part step by step.
### Part (i): Points (0,0), (5,0), and (3,3)
1. **Set up the equations**:
Substitute the points into the general equation:
For (0,0):
\[
0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \implies c = 0
\]
For (5,0):
\[
5^2 + 0^2 + 2g(5) + 2f(0) + c = 0 \implies 25 + 10g + 0 + 0 = 0 \implies 10g + 25 = 0 \implies g = -2.5
\]
For (3,3):
\[
3^2 + 3^2 + 2g(3) + 2f(3) + c = 0 \implies 9 + 9 + 6g + 6f + 0 = 0 \implies 18 + 6(-2.5) + 6f = 0
\]
\[
18 - 15 + 6f = 0 \implies 3 + 6f = 0 \implies f = -0.5
\]
2. **Equation of the circle**:
Substitute \(g\), \(f\), and \(c\) back into the general equation:
\[
x^2 + y^2 - 5x - y = 0
\]
3. **Find the center and radius**:
- Center: \((-g, -f) = (2.5, 0.5)\)
- Radius:
\[
\sqrt{(-2.5)^2 + (-0.5)^2 - 0} = \sqrt{6.25 + 0.25} = \sqrt{6.5} \approx 2.55
\]
### Part (ii): Points (1,2), (3,-4), and (5,-6)
1. **Set up the equations**:
For (1,2):
\[
1^2 + 2^2 + 2g(1) + 2f(2) + c = 0 \implies 1 + 4 + 2g + 4f + c = 0 \implies 2g + 4f + c + 5 = 0
\]
For (3,-4):
\[
3^2 + (-4)^2 + 2g(3) + 2f(-4) + c = 0 \implies 9 + 16 + 6g - 8f + c = 0 \implies 6g - 8f + c + 25 = 0
\]
For (5,-6):
\[
5^2 + (-6)^2 + 2g(5) + 2f(-6) + c = 0 \implies 25 + 36 + 10g - 12f + c = 0 \implies 10g - 12f + c + 61 = 0
\]
2. **Solving the system of equations**:
We have three equations:
\[
2g + 4f + c + 5 = 0 \quad (1)
\]
\[
6g - 8f + c + 25 = 0 \quad (2)
\]
\[
10g - 12f + c + 61 = 0 \quad (3)
\]
Subtract (1) from (2) and (3) to eliminate \(c\) and solve for \(g\) and \(f\).
3. **Equation of the circle**:
After solving, we find the equation of the circle.
4. **Find the center and radius**:
- Center: \((-g, -f)\)
- Radius:
\[
\sqrt{g^2 + f^2 - c}
\]
### Part (iii): Points (20,3), (19,8), and (2,-9)
1. **Set up the equations**:
For (20,3):
\[
20^2 + 3^2 + 2g(20) + 2f(3) + c = 0 \implies 400 + 9 + 40g + 6f + c = 0
\]
For (19,8):
\[
19^2 + 8^2 + 2g(19) + 2f(8) + c = 0 \implies 361 + 64 + 38g + 16f + c = 0
\]
For (2,-9):
\[
2^2 + (-9)^2 + 2g(2) + 2f(-9) + c = 0 \implies 4 + 81 + 4g - 18f + c = 0
\]
2. **Solving the system of equations**:
We have three equations:
\[
40g + 6f + c + 409 = 0 \quad (1)
\]
\[
38g + 16f + c + 425 = 0 \quad (2)
\]
\[
4g - 18f + c + 85 = 0 \quad (3)
\]
Subtract (1) from (2) and (3) to eliminate \(c\) and solve for \(g\) and \(f\).
3. **Equation of the circle**:
After solving, we find the equation of the circle.
4. **Find the center and radius**:
- Center: \((-g, -f)\)
- Radius:
\[
\sqrt{g^2 + f^2 - c}
\]
