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Point Slope Form of a Line

Find the equa...

Question

Find the equations to the straight lines passing through the pairs of points. : $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ .

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Solution

Given points are $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$
Then, equation of line is
$y - 2 a t_{1} = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}} (x - a t_{1}^{2})$
$\Rightarrow (y - 2 a t_{1}) \times (t_{1} + t_{2}) = 2 x - 2 a t_{1}^{2}$
$\Rightarrow y (t_{1} + t_{2}) - 2 a t_{1} (t_{1} + t_{2}) = 2 x - 2 a t_{1}^{2}$
$\Rightarrow y (t_{1} + t_{2}) - 2 a t_{1} t_{2} - 2 x = 0$ is the required equation.

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