Parabola

Any tangent to the parabola ${y^2} = 8ax$ is

$y = mx + {{2a} \over m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$

If $(i)$ is a tangent to the circle, ${x^2} + {y^2} = 2{a^2}$ then,

$\sqrt {2a} = \pm {{2a} \over {m\sqrt {{m^2} + 1} }}$

$ \Rightarrow {m^2}\left( {1 + {m^2}} \right) = 2$

$ \Rightarrow \left( {{m^2} + 2} \right)\left( {{m^2} - 1} \right) = 0$

$ \Rightarrow m = \pm 1.$

So from $(i),$ $y = \pm \left( {x + 2a} \right).$