Question

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

Answer 1

The distance d between two points and 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’.

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

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

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

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

Squaring on both sides of the equation we get,

$$\begin{gathered} {\left(x-5\right)}^2+{\left(y-1\right)}^2={\left(x-1\right)}^2+{\left(y-5\right)}^2 \\ \Rightarrow x^2+25-10x+y^2+1-2y=x^2+1-2x+y^2+25-10y \\ \Rightarrow 26-10x-2y=26-10y-2x \\ \Rightarrow 10y-2y=10x-2x \\ \Rightarrow 8y=8x \\ \Rightarrow y=x \end{gathered}$$

Hence we have proved that x = y.