Given : A circle, $$2x^2 + 2y^2 = 5$$ and a parabola, $$y^2 = 4\sqrt{5}x$$.
Statement - I : An equation of a common tangent to these curves is $$y = x + \sqrt{5}$$.
Statement - II : If the line, $$y = mx + \frac{\sqrt{5}}{m}$$ $$(m \neq 0)$$ is their common tangent, then $$m$$ satisfies $$m^4 - 3m^2 + 2 = 0$$.

Given Equations:

Circle: $$2x^2 + 2y^2 = 5 \implies x^2 + y^2 = \frac{5}{2}$$

Center $$(0, 0)$$ and radius $$r = \sqrt{\frac{5}{2}}$$.

Parabola: $$y^2 = 4\sqrt{5}x$$

Comparing with $$y^2 = 4ax$$, we get $$a = \sqrt{5}$$.

Condition for Common Tangent: The equation of any tangent to the parabola $$y^2 = 4ax$$ in slope form is:

$$y = mx + \frac{a}{m} \implies y = mx + \frac{\sqrt{5}}{m}$$

$$mx - y + \frac{\sqrt{5}}{m} = 0 \quad \text{--- (i)}$$

For this line to be tangent to the circle $$x^2 + y^2 = r^2$$, the perpendicular distance from the center $$(0, 0)$$ to the line must equal the radius $$r$$.

$$\left| \frac{m(0) - 0 + \frac{\sqrt{5}}{m}}{\sqrt{m^2 + (-1)^2}} \right| = \sqrt{\frac{5}{2}}$$

$$\frac{\sqrt{5}}{|m|\sqrt{m^2 + 1}} = \sqrt{\frac{5}{2}}$$

$$\frac{5}{m^2(m^2 + 1)} = \frac{5}{2}$$

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

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

Since $$m^2$$ cannot be negative, $$m^2 = 1 \implies m = \pm 1$$.

Thus, Statement - I is true.

Statement - II:

The $$m$$ values of the common tangents satisfy the equation given in Statement - II. Thus, Statement - II is true.

However, the actual condition for the common tangent is $$m^4 + m^2 - 2 = 0$$. Statement - II provides a different equation that happens to share the same roots ($$m^2=1$$), but it does not represent the logical derivation or the complete set of conditions for Statement - I. Therefore, it is not the correct explanation.