Step-by-step Solution:

Let common tangent to the curves be \[ y = mx + c \] \[ = mx + \frac{a}{m} \] Given the parabola equation: \[ y^2 = 8ax = 4(2a)x \] Equation of tangent to parabola: \[ y = mx + \frac{2a}{m} \quad \dots (2) \] which is also tangent to the circle: \[ x^2 + y^2 = 2a^2 = (\sqrt{2}a)^2 \] \( \textbf{Now Distance from (0,0) to the tangent line = Radius of circle} \) \[ \sqrt{2} a = \pm \frac{2a}{m} \times \frac{1}{\sqrt{1+m^2}} \] \[ \Rightarrow m^2(1 + m^2) - 2 = 0 \] \[ \Rightarrow (m^2 - 1)(m^2 + 2) = 0 \] \[ \Rightarrow m = \pm 1 \] \( \textbf{Required equation of tangent:} \) \[ y = mx + \frac{2a}{m} \] \[ \Rightarrow y = \pm (x + 2a) \]