Suppose that A5,6and 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 A(5,6) and B(3,4) 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-axis  The image of B(3,4) in the x-axis is D(3,−4) The equation of the line joining D(3,−4) and A(5,6) is y−6=6+45−3(x−5)This can be simplified as y−6=6+45−3x−5y−6=5x−5y−6=5x−255x−y−19=0Plug in  y=0 to get the point of intersection of the above line and  x-axis. It implies that5x=19x=195 Therefore, P=195,0