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

Given:

Three points $A( 0,\ 2)$, $B( 3,\ p)$ and $C( p,\ 5)$. Point $A$ is equidistant from the points $B$ and $C$.

To do:

We have to find the value of $p$.

Solution:

We know that,

The distance between the two points $(x_1, y_1)$ and $(x_2,y_2)=\sqrt{( x_{2} -x_{1})^{2} +( y_{2} -y_{1})^{2}}$

Therefore,

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

$\Rightarrow AB=\sqrt{9+( p-2)^{2}}$

Similarly,

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

$\Rightarrow AC=\sqrt{p^{2} +9}$

$AB=AC$

$\Rightarrow \sqrt{9+( p-2)^{2}} =\sqrt{p^{2} +9}$

Squaring on both sides, we get,

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

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

$\Rightarrow 4-4p=0$

$\Rightarrow 4p=4$

$\Rightarrow p=\frac{4}{4}$

$\Rightarrow p=1$

Therefore, the value of $p$ is $1$. 

Tutorialspoint

Updated on: 10-Oct-2022

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