Question

Find the slope of a line passing through the following point:

\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]

Answer 1

\[(a t_1^2 , 2 a t_1 ) \text { and } (a t_2^2 , 2 a t_2 )\]

Let m be the slope of the given line.

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

\[ \Rightarrow m = \frac{2a t_2 - 2a t_1}{a {t_2}^2 - a {t_1}^2} = \frac{2\left( t_2 - t_1 \right)}{\left( t_2 - t_1 \right)\left( t_2 + t_1 \right)} = \frac{2}{t_1 + t_2}\]

Hence, the slope of the line passing through the points

\[(a t_1^2 , 2a t_1 ) \text { and }(a t_2^2 , 2a t_2 )\] is \[\frac{2}{t_1 + t_2}\].