If the point P(x, y) is equidistant from the points


If the point P(xy) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.


The distance $d$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula


The three given points are P(x, y), A(5,1) and B(1,5).

Now let us find the distance between ‘P’ and ‘A’.

$P A=\sqrt{(x-5)^{2}+(y-1)^{2}}$

Now, let us find the distance between ‘P’ and ‘B’.

$P B=\sqrt{(x-1)^{2}+(y-5)^{2}}$

It is given that both these distances are equal. So, let us equate both the above equations,

PA = PB 


Squaring on both sides of the equation we get,


$\Rightarrow x^{2}+25-10 x+y^{2}+1-2 y=x^{2}+1-2 x+y^{2}+25-10 y$

$\Rightarrow 26-10 x-2 y=26-10 y-2 x$

$\Rightarrow 10 y-2 y=10 x-2 x$

$\Rightarrow 8 y=8 x$

$\Rightarrow y=x$

Hence we have proved that x y.


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