Find the slope of a line passing through the following points:
(i) (−3, 2) and (1, 4)
(ii) $(a t_{1}^{2}, 2 a t_{1}) and (a t_{2}^{2}, 2 a t_{2})$
(iii) (3, −5), and (1, 2)

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Solution

(i) (−3, 2) and (1, 4)

Let m be the slope of the given line.

$∴ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \Rightarrow m = \frac{4 - 2}{1 + 3} = \frac{2}{4} = \frac{1}{2}$

Hence, the slope of the line passing through the points (−3, 2) and (1, 4) is $\frac{1}{2}$ .

(ii) $(a t_{1}^{2}, 2 a t_{1}) and (a t_{2}^{2}, 2 a t_{2})$

Let m be the slope of the given line.

$∴ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \Rightarrow m = \frac{2 a t_{2} - 2 a t_{1}}{a {t_{2}}^{2} - a {t_{1}}^{2}} = \frac{2 (t_{2} - t_{1})}{(t_{2} - t_{1}) (t_{2} + t_{1})} = \frac{2}{t_{1} + t_{2}}$

Hence, the slope of the line passing through the points $(a t_{1}^{2}, 2 a t_{1}) and (a t_{2}^{2}, 2 a t_{2})$ is $\frac{2}{t_{1} + t_{2}}$ .

(iii) (3, −5), and (1, 2)

Let m be the slope of the given line.

$∴ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \Rightarrow m = \frac{2 + 5}{1 - 3} = - \frac{7}{2}$

Hence, the slope of the line passing through the points (3, −5), and (1, 2) is $- \frac{7}{2}$ .

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