Find the equations to the straight lines passing through the pair of points $(a t_{1}, \frac{a}{t_{1}})$ and $(a t_{2}, \frac{a}{t_{2}})$ .

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Solution

Line passing through points $A (x_{1}, y_{2}), B (x_{2}, y_{2})$ is given by
$y - y_{1} = (\frac{y_{2} - y_{1}}{x_{2} - x_{1}}) (x - x_{1})$
Equation of line joining $(a t_{1}, \frac{a}{t_{1}})$ and $(a t_{2}, \frac{a}{t_{2}})$ is

$y - \frac{a}{t_{1}} = \frac{(\frac{a}{t_{2}} - \frac{a}{t_{1}})}{a t_{2} - a t_{1}} (x - a t_{1}) \frac{t_{1} y - a}{t_{1}} = \frac{(\frac{a (t_{1} - t_{2})}{t_{1} t_{2}})}{a (t_{2} - t_{1})} (x - a t_{1}) \frac{t_{1} y - a}{t_{1}} = - \frac{1}{t_{1} t_{2}} (x - a t_{1}) (t_{1} y - a) t_{2} = - (x - a t_{1}) t_{1} t_{2} y - a t_{2} = - x + a t_{1} t_{1} t_{2} y + x = a (t_{1} + t_{2})$

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Similar questions

[a t_{1} t_{2}, a (t_{1} + t_{2})], [a t_{2} t_{3}, a (t_{2} + t_{3})], [a t_{3} t_{1}, a (t_{3} + t_{1})]

(a, b)

(a + b, a - b)

(3, 4)

(5, 6)

A (a t_{1} t_{2}, a (t_{1} + t_{2})), B (a t_{2} t_{3}, a (t_{2} + t_{3}))

C (a t_{3} t_{1}, a (t_{3} + t_{1}))

A B C

t_{1} \neq t_{2} \neq t_{3})

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