Question

Find the equation of the straight lines passing through the following pair of point :

(at₁, a/t₁) and (at₂, a/t₂)

Answer 1

(at₁, a/t₁) and (at₂, a/t₂)

\[\text { Here }, \left( x_1 , y_1 \right) \equiv \left( a t_1 , \frac{a}{t_1} \right) \]

\[\left( x_2 , y_2 \right) \equiv \left( a t_2 , \frac{a}{t_2} \right)\]

So, the equation of the line passing through the two points is

\[y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}\left( x - x_1 \right)\]

\[ \Rightarrow y - \frac{a}{t_1} = \frac{\frac{a}{t_2} - \frac{a}{t_1}}{a t_2 - a t_1}\left( x - a t_1 \right)\]

\[ \Rightarrow y - \frac{a}{t_1} = \frac{- 1}{t_2 t_1}\left( x - a t_1 \right)\]

\[ \Rightarrow x + t_1 t_2 y = a\left( t_1 + t_2 \right)\]