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Angle between Two Lines

lf the line j...

Question

lf the line joining the points $(a t_{1}^{2}, 2 a t_{1}), (a t_{2}^{2}, 2 a t_{2})$ is parallel to $y = x,$ then $t_{1} + t_{2} =$

A

\frac{1}{2}

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B

4

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C

\frac{1}{4}

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D

2

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Solution

The correct option is D $2$
Slope of line joining the points $(a t_{1}^{2}, 2 a t_{1}), (a t_{2}^{2}, 2 a t_{2})$ is $m = \frac{2 a (t_{2} - t_{1})}{a (t_{2}^{2} - t_{1}^{2})}$
$\Rightarrow m = \frac{2}{(t_{1} + t_{2})}$
Now, given line is $y = x$
Slope of given line is $1.$
Since,the line joining given points is parallel to $y = x$
So, their slopes are equal
$\frac{2}{t_{1} + t_{2}} = 1$
$t_{1} + t_{2} = 2$

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