The differential equation of the family of curves ${\mathrm{r}}^2 = {\mathrm{a}}^2 \cos 2\mathrm{\theta }$ where ‘a’ is an arbitary constant is :

$\begin{array}{l}Given{r}^2=a^2\cos 2\theta --\left(1\right)\\ \Rightarrow 2r\frac{dr}{d\theta }=a^2-2\sin 2\theta \\ \Rightarrow r\frac{dr}{d\theta }=-a^2\sin 2\theta \\ \Rightarrow a^2=\frac{-r.\left(\frac{dr}{d\theta }\right)}{\sin 2\theta }\end{array}$

$\begin{array}{l}From\left(1\right)\\ \therefore r^2=\frac{-r\left(\frac{dr}{d\theta }\right)}{\sin 2\theta }.\cos 2\theta \\ r\sin 2\theta =-\left(\frac{dr}{d\theta }\right)\cos 2\theta \\ \Rightarrow \cos 2\theta \left(\frac{dr}{d\theta }\right)+r\sin 2\theta =0\end{array}$