Suppose that A5,6 and B3,4 be two points and P  be a point on  x- axis such that PA+PB is minimum then the point P is

Suppose that  the images of A5,6 and B3,4 in the X-  axis are  C and D respectively. The point P  is the point of intersection of the lines  AD and  x- axis, it is coincide with the point of intersection of the lines  BC and x-  axisThe image of B3,4 in the x-  axis is  D3,−4The equation of the line joining D3,−4 an A5,6 is  y−6=6+45−3x−5This can be simplified as y−6=6+45−3x−5y−6=5x−5y−6=5x−255x−y−19=0Plug in  x=0 to get the point of intersection of the above line and    axis. It implies that5x=19x=195Therefore, P=195,0