State whether the two lines in the following problems are parallel, perpendicular or neither parallel nor perpendicular :
`(i)` Through `(5,6)` and `(2,3)` , through `(9,-2)` and `(6,-5)`
`(ii)` Through `(2,-5)` and `(-2,5)`, through `(6,3)` and `(1,1)`.

Answer

To determine whether the two lines in each of the given problems are parallel, perpendicular, or neither, we will calculate the slopes of the lines formed by the given points.

### Problem (i):
**Points:** (5, 6) and (2, 3), (9, -2) and (6, -5)

1. **Calculate the slope of the first line (m1):**
- Using the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
- Let \( (x_1, y_1) = (5, 6) \) and \( (x_2, y_2) = (2, 3) \).
- \( m_1 = \frac{3 - 6}{2 - 5} = \frac{-3}{-3} = 1 \)

2. **Calculate the slope of the second line (m2):**
- Let \( (x_1, y_1) = (9, -2) \) and \( (x_2, y_2) = (6, -5) \).
- \( m_2 = \frac{-5 + 2}{6 - 9} = \frac{-3}{-3} = 1 \)

3. **Compare the slopes:**
- Since \( m_1 = m_2 = 1 \), the lines are **parallel**.

### Problem (ii):
**Points:** (2, -5) and (-2, 5), (6, 3) and (1, 1)

1. **Calculate the slope of the first line (m1):**
- Let \( (x_1, y_1) = (2, -5) \) and \( (x_2, y_2) = (-2, 5) \).
- \( m_1 = \frac{5 - (-5)}{-2 - 2} = \frac{10}{-4} = -\frac{5}{2} \)

2. **Calculate the slope of the second line (m2):**
- Let \( (x_1, y_1) = (6, 3) \) and \( (x_2, y_2) = (1, 1) \).
- \( m_2 = \frac{1 - 3}{1 - 6} = \frac{-2}{-5} = \frac{2}{5} \)

3. **Compare the slopes:**
- To check if the lines are perpendicular, we multiply the slopes:
- \( m_1 \cdot m_2 = -\frac{5}{2} \cdot \frac{2}{5} = -1 \)
- Since the product of the slopes is -1, the lines are **perpendicular**.

### Summary of Results:
- **(i)** The lines are **parallel**.
- **(ii)** The lines are **perpendicular**.

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