State whether the two lines in each of the following problems are parallel , perpendicular or neither parallel nor perpendicular :
`(i)` Through `(8,2)` and `(-5,3)` , through `(16,6)` and `(3,15)`
`(ii)` Through `(9,5)` and `(-1,1)`, through `(8,-3)` and `(3,-5)`
`(iii)` Through `(-2,6)` and `(4,8)` , through `(8,12)` and `(4,24)`.

Answer

To determine whether the two lines in each part of the question are parallel, perpendicular, or neither, we will calculate the slopes of the lines using the formula for slope \( m \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Part (i)

**Lines through points (8,2) and (-5,3) and through (16,6) and (3,15)**

1. **Calculate the slope \( m_1 \) for the first line:**
- Points: \( (8, 2) \) and \( (-5, 3) \)
- \( m_1 = \frac{3 - 2}{-5 - 8} = \frac{1}{-13} = -\frac{1}{13} \)

2. **Calculate the slope \( m_2 \) for the second line:**
- Points: \( (16, 6) \) and \( (3, 15) \)
- \( m_2 = \frac{15 - 6}{3 - 16} = \frac{9}{-13} = -\frac{9}{13} \)

3. **Check the conditions:**
- \( m_1 \neq m_2 \) (since \(-\frac{1}{13} \neq -\frac{9}{13}\))
- \( m_1 \cdot m_2 = -\frac{1}{13} \cdot -\frac{9}{13} = \frac{9}{169} \neq -1 \)

**Conclusion for Part (i):** The lines are neither parallel nor perpendicular.

### Part (ii)

**Lines through points (9,5) and (-1,1) and through (8,-3) and (3,-5)**

1. **Calculate the slope \( m_1 \) for the first line:**
- Points: \( (9, 5) \) and \( (-1, 1) \)
- \( m_1 = \frac{1 - 5}{-1 - 9} = \frac{-4}{-10} = \frac{2}{5} \)

2. **Calculate the slope \( m_2 \) for the second line:**
- Points: \( (8, -3) \) and \( (3, -5) \)
- \( m_2 = \frac{-5 + 3}{3 - 8} = \frac{-2}{-5} = \frac{2}{5} \)

3. **Check the conditions:**
- \( m_1 = m_2 \) (since \(\frac{2}{5} = \frac{2}{5}\))

**Conclusion for Part (ii):** The lines are parallel.

### Part (iii)

**Lines through points (-2,6) and (4,8) and through (8,12) and (4,24)**

1. **Calculate the slope \( m_1 \) for the first line:**
- Points: \( (-2, 6) \) and \( (4, 8) \)
- \( m_1 = \frac{8 - 6}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \)

2. **Calculate the slope \( m_2 \) for the second line:**
- Points: \( (8, 12) \) and \( (4, 24) \)
- \( m_2 = \frac{24 - 12}{4 - 8} = \frac{12}{-4} = -3 \)

3. **Check the conditions:**
- \( m_1 \neq m_2 \) (since \(\frac{1}{3} \neq -3\))
- \( m_1 \cdot m_2 = \frac{1}{3} \cdot -3 = -1 \)

**Conclusion for Part (iii):** The lines are perpendicular.

### Summary of Results:
- Part (i): Neither parallel nor perpendicular.
- Part (ii): Parallel.
- Part (iii): Perpendicular.