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Distance between Two Points on the Same Coordinate Axes

Find the dist...

Question

Find the distance between the points
$A ({a t}_{1}^{2}, {2 a t}_{1})$ and $B ({a t}_{2}^{2}, {2 a t}_{2})$

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Solution

We have,
$A B = \sqrt{{({a t}_{2}^{2} - {a t}_{1}^{2})}^{2} + {({2 a t}_{2} - {2 a t}_{1})}^{2}}$
$\Rightarrow A B \sqrt{a^{2} {(t_{2} - t_{1})}^{2} {(t_{2} + t_{1})}^{2} + {4 a}^{2} {(t_{2} - t_{1})}^{2}}$
$\Rightarrow A B = a (t_{2} - t_{1}) \sqrt{{(t_{2} + t_{1})}^{2} + 4}$

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