According to the question we have
\(A(2,-5)\) and \(B(-2,9)\)
Let the points be \(P(x, 0)\).
So, \(A P=P B\) and \(A P^{2}=P B^{2}\)
\(=(x-2)^{2}-(0+5)^{2}=(x+2)^{2}+(0-9)^{2}\)
\(=x^{2}+4-4 x+25=x^{2}+4+4 x-81\)
\(\Rightarrow x^{2}+29-4 x=x^{2}+85+4 x\)
\(=-4 x-4 x=85-29\)
\(=-8 x=56\)
\(\Rightarrow x=-7\)
Hence, point on the \(x\)-axis which is equidistant from \((2,-5)\) and \((-2,9)\) is \((-7,0) .\)
