State whether the two lines in each of the following are parallel, perpendicular or neither.
(i) Through (5, 6) and (2, 3); through (9, −2) and (6, −5)
(ii) Through (9, 5) and (−1, 1); through (3, −5) and (8, −3)
(iii) Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)
(iv) Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).

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Solution

(i) Through (5, 6) and (2, 3); through (9, −2) and (6, −5)

Let m₁ be the slope of the line joining (5, 6) and (2, 3) and m₂ be the slope of the line joining (9, −2) and (6, −5).

$∴ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{3 - 6}{2 - 5} = \frac{- 3}{- 3} = 1 and m_{2} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 5 + 2}{6 - 9} = \frac{- 3}{- 3} = 1$

$Since, m_{1} = m_{2}$

Therefore, the given lines are parallel.

(ii) Through (9, 5) and (−1, 1); through (3, −5) and (8, −3)

Let m₁ be the slope of the line joining (9, 5) and (−1, 1) and m₂ be the slope of the line joining (3, −5) and (8, −3).

$∴ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{1 - 5}{- 1 - 9} = \frac{- 4}{- 10} = \frac{2}{5} and m_{2} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 3 + 5}{8 - 3} = \frac{2}{5}$

$Since, m_{1} = m_{2}$

Therefore, the given lines are parallel.

(iii) Through (6, 3) and (1, 1); through (−2, 5) and (2, −5).

Let m₁ be the slope of the line joining (6, 3) and (1, 1) and m₂ be the slope of the line joining (−2, 5) and (2, −5).

$∴ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{1 - 3}{1 - 6} = \frac{- 2}{- 5} = \frac{2}{5} and m_{2} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 5 - 5}{2 + 2} = \frac{- 10}{4} = \frac{- 5}{2}$

$Now, m_{1} m_{2} = \frac{2}{5} \times \frac{- 5}{2} = - 1 Since, m_{1} m_{2} = - 1$

Therefore, the given lines are perpendicular.

(iv) Through (3, 15) and (16, 6); through (−5, 3) and (8, 2).

Let m₁ be the slope of the line joining (3, 15) and (16, 6) and m₂ be the slope of the line joining (−5, 3) and (8, 2).

$∴ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{6 - 15}{16 - 3} = \frac{- 9}{13} and m_{2} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{2 - 3}{8 + 5} = \frac{- 1}{13}$

$Now, m_{1} m_{2} = \frac{- 9}{13} \times \frac{- 1}{13} = \frac{9}{169} Since, m_{1} m_{2} \neq - 1 and m_{1} \neq m_{2}$

Therefore, the given lines are neither parallel nor perpendicular.

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