Equation of common tangent to ${\mathrm{x}}^2+{\mathrm{y}}^2=2{\mathrm{a}}^2, {\mathrm{y}}^2=8\mathrm{ax}$ is

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2=2{\mathrm{a}}^2 ; {\mathrm{y}}^2=8\mathrm{ax} \\ \mathrm{c}=\left(0,0\right), \mathrm{r}=\sqrt{2}\mathrm{a} ; \\ {\mathrm{y}}^2=8\mathrm{ax} \\ {\mathrm{y}}^2=4Ax \\ \Rightarrow 4\mathrm{A}=8\mathrm{a} \\ \Rightarrow \mathrm{A}=2\mathrm{a} \end{gathered}$$

Eq. of tangent to parabola

 $$\begin{gathered} \mathrm{y}=\mathrm{mx}+\mathrm{A}/\mathrm{m} \\ \mathrm{y}=\mathrm{mx}+2\mathrm{a}/\mathrm{m} \\ \mathrm{mx}-\mathrm{y}+2\mathrm{a}/\mathrm{m}=0 \end{gathered}$$

Common tangent to circle 

$\mathrm{r}=\mathrm{d}$

$$\begin{gathered} \sqrt{2}\mathrm{a}=\left|2\mathrm{a}/\mathrm{m}\right|/{\mathrm{m}}^2+1 \\ 2{\mathrm{a}}^2\left({\mathrm{m}}^2+1\right)=4{\mathrm{a}}^2/{\mathrm{m}}^2 \\ {\mathrm{m}}^2\left({\mathrm{m}}^2+1\right)=1\left(1+1\right) \\ {\mathrm{m}}^2=1 \\ \mathrm{m}=\pm 1 \end{gathered}$$

Eq. of tangent is 

$$\begin{gathered} \mathrm{y}=\pm \mathrm{x}\pm 2\mathrm{a} \\ \mathrm{y}=\pm \left(\mathrm{x}+2\mathrm{a}\right) \end{gathered}$$