Find the point on the x-axis which is equidistant from the points (2, -5) and (-2, 9).

We know that,

y-coordinate of any point on x-axis = 0.

By distance formula,

d = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

Let point on x-axis which is equidistant from the points (2, -5) and (-2, 9) be P(x, 0).

∴ Distance between (2, -5) and (x, 0) = Distance between (-2, 9) and (x, 0).

$\Rightarrow \sqrt{(x - 2)^2 + [0 - (-5)]^2} = \sqrt{[x - (-2)]^2 + (0 - 9)^2} \\[1em] \Rightarrow \sqrt{x^2 + 4 - 4x + 5^2} = \sqrt{(x + 2)^2 + (-9)^2} \\[1em] \Rightarrow \sqrt{x^2 - 4x + 25 + 4} = \sqrt{x^2 + 4 + 4x + 81} \\[1em]$

On squaring both sides,

$\Rightarrow x^2 - 4x + 29 = x^2 + 4x + 85 \\[1em] \Rightarrow x^2 - x^2 + 4x + 4x = 29 - 85 \\[1em] \Rightarrow 8x = -56 \\[1em] \Rightarrow x = -7.$

P = (x, 0) = (-7, 0).

Hence, required point is (-7, 0).