A straight line touches the circle  $x^2+y^2=2a^2$ and the parabola $y^2=8ax$. The equation of the line is

equation of the parabola is  $y^2=8ax$ ……….(1)

any tangent to it is given by $y=mx+\frac{2a}{m}\mathrm{...........}\left(2\right)$

since it touches the circle  $x^2+y^2=2a^2$

$\therefore$the length of $\perp$from centre (0, 0) to (2) is equal to the radius $\sqrt{2}a$

$\Rightarrow \frac{{\displaystyle \frac{2a}{m}}}{\sqrt{1+m^2}}=\pm \sqrt{2}a$$\Rightarrow \frac{2a}{m}=\pm \sqrt{2}a\sqrt{1+m^2}$

Squaring,  $\frac{4}{m^2}=2(1+m^2)$

or $m^4+m^2-2=0$ or $(m^2+2)(m^2-1)=0$ 

$\therefore$ $m^2=-2,1\Rightarrow m^2=1orm=\pm 1$

Hence from (2),  $y=\pm x\pm 2a$ i.e. $y=\pm (x+2a)$ 

Which are the required equations of common tangents