Chapter 10.2, Problem 75ES

In the late seventeenth century the Italian astronomer Giovanni Domenico Cassini (1625-1712) introduced the family of curves ${\left(x^2+y^2+a^2\right)}^2-b^4-4a^2{x}^2=0\left(a>0,b>0\right)$ in his studies of the relative motions of the Earth and the Sun. These curves, which are called Cassini ovals, have one of the three basic shapes shown in the accompanying figure.

(a) Show that if $a=b,$ then the polar equation of the Cassini oval is $r^2=2a^2\cos 2\theta ,$ which is a lemniscate.

(b) Use the formula in Exercise 71 to show that the lemniscate in part (a) is the curve traced by a point that moves in such a way that the product of its distances from the polar points $\left(a,0\right)\text{ and }\left(a,\pi \right)\text{ is }{a}^2.$