Let M be the point on X-axis which is equidistant from P(2,-5) and Q(-2,9).

Since the point M is on X-axis, its y co-ordinate is zero.

M = (x,0)

Since M is equidistant from P and Q,

PM = QM ………..(i)

by Distance formula, PM = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]

= √[(x-2)^{2}+(0-(-5))^{2}]

= √[(x-2)^{2}+(5)^{2}]

= √(x^{2}-4x+4+25)

= √(x^{2}-4x+29)

by Distance formula, QM = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]

= √[(x-(-2))^{2}+(0-9)^{2}]

= √[(x+2)^{2}+(9)^{2}]

= √(x^{2}+4x+85)

From (i)

√(x^{2}-4x+29) = √(x^{2}+4x+85)

Squaring both sides

x^{2}-4x+29 = x^{2}+4x+85

-8x = 85-29

-8x = 56

x = 56/-8

x = -7

Hence the point on X axis equidistant from P(2,-5) and Q(-2,9) is (-7,0).