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Distance Formula

If t 1 and t ...

Question

If $t_{1}$ and $t_{2}$ are roots of the equation $t^{2} - 2 \sqrt{3} t + 2 = 0$ , then the distance between the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ , where $a > 0$ is

A

2 a units

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B

a units

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C

4 a units

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D

8 a units

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Solution

The correct option is D $8 a units$
Given:
$t^{2} - 2 \sqrt{3} t + 2 = 0$
From sum and product of roots, we get
$t_{1} + t_{2} = 2 \sqrt{3} t_{1} t_{2} = 2$
Now, the distance between the given points
$= \sqrt{(a t_{1}^{2} - a t_{2}^{2})^{2} + (2 a t_{1} - 2 a t_{2})^{2}} = a | t_{1} - t_{2} | \sqrt{(t_{1} + t_{2})^{2} + 4} = a \sqrt{(t_{1} + t_{2})^{2} - 4 t_{1} t_{2}} \sqrt{12 + 4} = 4 a \sqrt{12 - 8} = 8 a units$

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