Find the point on the X-axis which is equidistant from A(-3, 4) and B(1, -4).

Let the point on the x-axis be P(a, 0).

\[PA = \sqrt{\left( a - \left( - 3 \right) \right)^2 + \left( 0 - 4 \right)^2}\]

\[ = \sqrt{\left( a + 3 \right)^2 + 16}\]

\[PB = \sqrt{\left( a - 1 \right)^2 + \left( 0 - \left( - 4 \right) \right)^2}\]

\[ = \sqrt{\left( a - 1 \right)^2 + 16}\]

\[ {PA}^2 = {PB}^2 \]

\[ \Rightarrow \left( a + 3 \right)^2 + 16 = \left( a - 1 \right)^2 + 16\]

\[ \Rightarrow a^2 + 6a + 9 = a^2 - 2a + 1\]

\[ \Rightarrow 8a = - 8\]

\[ \Rightarrow a = - 1\]

\[\left( a, 0 \right) = \left( - 1, 0 \right)\]