Question

Find the point on x-axis that is equidistant from the points A(−3, 2) and B(5, −2).

Answer 1

Let the point P(x,0) be equidistant from the points $A\left(-3,2\right)\text{and}B\left(5,-2\right)$.
Then,
$$\begin{gathered} PA=PB \\ \Rightarrow P{A}^2=P{B}^2 \\ \Rightarrow {\left(x+3\right)}^2+{\left(0-2\right)}^2={\left(x-5\right)}^2+{\left(0+2\right)}^2 \\ \Rightarrow x^2+6x+13=x^2-10x+29 \\ \Rightarrow 16x=16 \\ \Rightarrow x=1 \end{gathered}$$
Therefore,
$P\left(x,0\right)\equiv P\left(1,0\right)$