Find the equations of the circle for which the points given below are the end points of a diameter.
(i) (-4,3), (3,-4)
(ii) (7,-3), (3,5)
(iii)( 1,1), (2,-1)
(iv) (0,0),(8,5)
(v) (3,1),(2,7)

Answer

To find the equations of the circles for which the given points are the endpoints of a diameter, we can use the general formula for the equation of a circle based on its diameter endpoints \((x_1, y_1)\) and \((x_2, y_2)\):

\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

This can be expanded and simplified to derive the standard form of the circle's equation.

### Step-by-Step Solutions

#### (i) For points (-4, 3) and (3, -4):

1. **Identify the points**:
- \( (x_1, y_1) = (-4, 3) \)
- \( (x_2, y_2) = (3, -4) \)

2. **Substitute into the formula**:
\[ (x + 4)(x - 3) + (y - 3)(y + 4) = 0 \]

3. **Expand the equation**:
\[ x^2 + 4x - 3x - 12 + y^2 + 4y - 3y - 12 = 0 \]
\[ x^2 + y^2 + x + y - 24 = 0 \]

4. **Final equation**:
\[ x^2 + y^2 + x + y - 24 = 0 \]

#### (ii) For points (7, -3) and (3, 5):

1. **Identify the points**:
- \( (x_1, y_1) = (7, -3) \)
- \( (x_2, y_2) = (3, 5) \)

2. **Substitute into the formula**:
\[ (x - 7)(x - 3) + (y + 3)(y - 5) = 0 \]

3. **Expand the equation**:
\[ x^2 - 3x - 7x + 21 + y^2 - 5y + 3y - 15 = 0 \]
\[ x^2 + y^2 - 10x - 2y + 6 = 0 \]

4. **Final equation**:
\[ x^2 + y^2 - 10x - 2y + 6 = 0 \]

#### (iii) For points (1, 1) and (2, -1):

1. **Identify the points**:
- \( (x_1, y_1) = (1, 1) \)
- \( (x_2, y_2) = (2, -1) \)

2. **Substitute into the formula**:
\[ (x - 1)(x - 2) + (y - 1)(y + 1) = 0 \]

3. **Expand the equation**:
\[ x^2 - 2x - x + 2 + y^2 - 1 = 0 \]
\[ x^2 + y^2 - 3x + 1 = 0 \]

4. **Final equation**:
\[ x^2 + y^2 - 3x + 1 = 0 \]

#### (iv) For points (0, 0) and (8, 5):

1. **Identify the points**:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (8, 5) \)

2. **Substitute into the formula**:
\[ (x - 0)(x - 8) + (y - 0)(y - 5) = 0 \]

3. **Expand the equation**:
\[ x^2 - 8x + y^2 - 5y = 0 \]

4. **Final equation**:
\[ x^2 + y^2 - 8x - 5y = 0 \]

#### (v) For points (3, 1) and (2, 7):

1. **Identify the points**:
- \( (x_1, y_1) = (3, 1) \)
- \( (x_2, y_2) = (2, 7) \)

2. **Substitute into the formula**:
\[ (x - 3)(x - 2) + (y - 1)(y - 7) = 0 \]

3. **Expand the equation**:
\[ x^2 - 2x - 3x + 6 + y^2 - 7y + y - 7 = 0 \]
\[ x^2 + y^2 - 5x - 6y - 1 = 0 \]

4. **Final equation**:
\[ x^2 + y^2 - 5x - 6y - 1 = 0 \]