Question

If P(x, y) is a point equidistant from the points A(6, −1) and B(2, 3), show that x − y = 3.

Answer 1

The given points are A(6, −1) and B(2, 3). The point P(x, y) is equidistant from the points A and B. So, PA = PB.
Also, (PA)² = (PB)^{2
$$\begin{gathered} \Rightarrow {\left(6-x\right)}^2+{\left(-1-y\right)}^2={\left(2-x\right)}^2+{\left(3-y\right)}^2 \\ \Rightarrow x^2-12x+36+y^2+2y+1=x^2-4x+4+y^2-6y+9 \\ \Rightarrow x^2+y^2-12x+2y+37=x^2+y^2-4x-6y+13 \\ \Rightarrow x^2+y^2-12x+2y-x^2-y^2+4x+6y=13-37 \\ \Rightarrow -8x+8y=-24 \\ \Rightarrow -8\left(x-y\right)=-24 \\ \Rightarrow x-y=\frac{-24}{-8} \\ \Rightarrow x-y=3 \end{gathered}$$}
Hence proved.