To solve the given problems step by step, we will break down each part of the question.
### Part (i): Find the other end of the diameter of the circle given one end (2, 3).
1. **Identify the equation of the circle**:
The equation of the circle is given as:
\[
x^2 + y^2 - 8x - 8y + 27 = 0
\]
2. **Rewrite the equation in standard form**:
To find the center of the circle, we will complete the square.
\[
(x^2 - 8x) + (y^2 - 8y) + 27 = 0
\]
Completing the square:
\[
(x - 4)^2 - 16 + (y - 4)^2 - 16 + 27 = 0
\]
\[
(x - 4)^2 + (y - 4)^2 - 5 = 0
\]
\[
(x - 4)^2 + (y - 4)^2 = 5
\]
The center of the circle is \(C(4, 4)\).
3. **Use the midpoint formula**:
Let the other end of the diameter be \(B(h, k)\). The midpoint \(M\) of the diameter \(AB\) is the center \(C\).
\[
M = \left(\frac{2 + h}{2}, \frac{3 + k}{2}\right) = (4, 4)
\]
4. **Set up equations**:
From the midpoint formula, we have:
\[
\frac{2 + h}{2} = 4 \quad \text{and} \quad \frac{3 + k}{2} = 4
\]
5. **Solve for \(h\) and \(k\)**:
- For the first equation:
\[
2 + h = 8 \implies h = 6
\]
- For the second equation:
\[
3 + k = 8 \implies k = 5
\]
6. **Conclusion**:
The other end of the diameter is \((6, 5)\).
### Part (ii): Show that \(A(3, -1)\) lies on the circle and find the other end of the diameter through \(A\).
1. **Identify the equation of the circle**:
The equation of the circle is:
\[
x^2 + y^2 - 2x + 4y = 0
\]
2. **Substitute point \(A(3, -1)\)**:
Substitute \(x = 3\) and \(y = -1\) into the equation:
\[
3^2 + (-1)^2 - 2(3) + 4(-1) = 0
\]
\[
9 + 1 - 6 - 4 = 0
\]
\[
0 = 0
\]
Thus, point \(A(3, -1)\) lies on the circle.
3. **Find the center of the circle**:
Rewrite the equation in standard form:
\[
(x^2 - 2x) + (y^2 + 4y) = 0
\]
Completing the square:
\[
(x - 1)^2 - 1 + (y + 2)^2 - 4 = 0
\]
\[
(x - 1)^2 + (y + 2)^2 = 5
\]
The center of the circle is \(C(1, -2)\).
4. **Use the midpoint formula**:
Let the other end of the diameter be \(B(x, y)\). The midpoint \(M\) of the diameter \(AB\) is the center \(C\).
\[
M = \left(\frac{3 + x}{2}, \frac{-1 + y}{2}\right) = (1, -2)
\]
5. **Set up equations**:
From the midpoint formula, we have:
\[
\frac{3 + x}{2} = 1 \quad \text{and} \quad \frac{-1 + y}{2} = -2
\]
6. **Solve for \(x\) and \(y\)**:
- For the first equation:
\[
3 + x = 2 \implies x = -1
\]
- For the second equation:
\[
-1 + y = -4 \implies y = -3
\]
7. **Conclusion**:
The other end of the diameter through \(A(3, -1)\) is \((-1, -3)\).
