If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find the value of p.


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


As per the question

$A B=A C$

$\Rightarrow \sqrt{(0-3)^{2}+(2-p)^{2}}=\sqrt{(0-p)^{2}+(2-5)^{2}}$

$\Rightarrow \sqrt{(-3)^{2}+(2-p)^{2}}=\sqrt{(-p)^{2}+(-3)^{2}}$

Squaring both sides, we get


$\Rightarrow 9+4+p^{2}-4 p=p^{2}+9$

$\Rightarrow 4 p=4$

$\Rightarrow p=1$


$A B=\sqrt{(0-3)^{2}+(2-p)^{2}}$

$=\sqrt{(-3)^{2}+(2-1)^{2}} \quad(\because p=1)$


$=\sqrt{10}$ units

Hence, $p=1$ and $A B=\sqrt{10}$ units.

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