The shape of the interface pictured in Fig. P.5.1 is known as a Cartesian oval after Ren̩ Descartes, who studied it in the 1600s. It۪s the perfect configuration to carry any ray from S to the interface to P. Prove that the defining equation is

Show that this is equivalent to

where x and y are the coordinates of point A.

Fig. P.5.1

constant loni + l;n2 n1(x + y)/2 + n2[y + ($, + s x)]'/2 = constant