The equation of the director circle of ${\mathrm{x}}^2/12–{\mathrm{y}}^2/8=1$ is

$$\begin{gathered} \frac{{\mathrm{x}}^2}{12}-\frac{{\mathrm{y}}^2}{8}=1 \\ \Rightarrow {\mathrm{a}}^2=12, {\mathrm{b}}^2=8 \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{Equation} \mathrm{of} \mathrm{director} \mathrm{circle} \mathrm{is} {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2-{\mathrm{b}}^2 \\ \Rightarrow {\mathrm{x}}^2 +{\mathrm{y}}^2=4 \end{gathered}$$