Suppose that A1,2 and B3,4 be two points and P  be a point on  y=x such that PA+PB is minimum then the  point  P

Suppose that the images of A(1,2) and B(3,4) in the line x=y are C and D respectively.  The point P is the point of intersection of the lines AD and x=y, it is coincide with the  point of intersection of the lines BC and x=y The image of B(3,4) in the line y=x is D(4,3) The equation of the line joining D(4,3) an A(1,2) is y−2=2−31−4(x−1)This can be simplified as y−2=13x−13y−6=x−1x−3y+5=0Plug in x=y  to get the point of intersection of the above line and the line x=y . It implies thatx−3x+5=02x=5x=52 Therefore, P=52,52