Find the point on the x-axis which is equidistant from (2, −5) and (−2, 9).
(−7, 0)
Let the required point on the x-axis be \((x, 0)\). A point on the x-axis has y–coordinate 0.
This point must be equidistant from \((2, -5)\) and \((-2, 9)\). So we set their distances equal.
Distance from \((x, 0)\) to \((2, -5)\): \( \sqrt{(x - 2)^2 + (0 + 5)^2} \).
Distance from \((x, 0)\) to \((-2, 9)\): \( \sqrt{(x + 2)^2 + (0 - 9)^2} \).
Equating the two distances (squaring both sides to remove the square roots):
\((x - 2)^2 + 25 = (x + 2)^2 + 81\).
Expand both sides: \( x^2 - 4x + 4 + 25 = x^2 + 4x + 4 + 81 \).
Simplify: \( x^2 - 4x + 29 = x^2 + 4x + 85 \).
Cancel \(x^2\): \( -4x + 29 = 4x + 85 \).
Bring variables together: \( -4x - 4x = 85 - 29 \).
So, \( -8x = 56 \).
Thus, \( x = -7 \).
The required point on the x-axis is \((-7, 0)\).