Show, using the focus-directrix property for an ellipse, that PS +PS'=2a where P is a point on the ellipse and S and S' are the two foci.

The focus diretrix property for an ellipse is PS/PD=e. Now this is also the case for the other directrix and focus, so PS'/PD'=e. Now we can rearrange these equations to find a formula for PS +PS', PS +PS'=e(PD+PD'). Using a simple diagram we can see PD+PD'= 2ae by marking on some distances. Subbing into the above equation yields PS +PS'=2a.