Question

If the point P(x,y) is equidistant from the points
A(a-b,a+b) and B(-a-b,a+b) , prove that x-a=0.

Answer 1

Are the coordinates of the points  A & B given correctly?

If a point P(x,y) is equidistant from two points A & B, then P lies on the perpendicular bisector of  AB.

We see that  y coordinates of A & B are the equal. So AB parallel to x axis. So perpendicular bisector is perpendicular to x axis. 
It is given by:  x = [(a-b) +  (-a-b) ] /2
                       x = -b   or   x + b = 0