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

Question:

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.

Solution:

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

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

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

$\Rightarrow 4 p=4$

$\Rightarrow p=1$

Now,

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

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

$=\sqrt{9+1}$

$=\sqrt{10}$ units

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

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