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

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

$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$

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 

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

Squaring on both sides of the equation we get,

$(x-5)^{2}+(y-1)^{2}=(x-1)^{2}+(y-5)^{2}$

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

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

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

$\Rightarrow 8 y=8 x$

$\Rightarrow y=x$

Hence we have proved that x = y.