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Question

If the point $$A(0,2)$$ is equidistant from the points $$B(3,p)$$ and $$C(p,5)$$, find $$\text p$$. Also, find the length of $$AB$$.


Solution

Given,
$$AB=AC$$
$$(AB)^{2} = (AC)^{2} $$

Distance between two points=$$\sqrt { { (x_2-x_1) }^{ 2 }+{ (y_2-y_1) }^{ 2 } } $$

$$(AB)^{2} = (AC)^{2} \implies$$$$(0-3)^{2}+(2-p)^{2} = (0-p)^{2} +(2-5)^{2}$$

$$9+4+p^{2}-4p=p^{2}+9$$
$$ p= 1$$

$$ Distance = \sqrt {(0-3)^{2} + (2-1)^{2}}$$
$$ Distance = \sqrt {10} $$

Mathematics

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