Statement 1: An equation of a common tangent to the parabola $$y^2 = 16\sqrt{3}x$$ and the ellipse $$2x^2 + y^2 = 4$$ is $$y = 2x + 2\sqrt{3}$$. 

Statement 2: If the line $$y = mx + \frac{4\sqrt{3}}{m}$$, $$(m \neq 0)$$ is a common tangent to the parabola $$y^2 = 16\sqrt{3}x$$ and the ellipse $$2x^2 + y^2 = 4$$, then $$m$$ satisfies $$m^4 + 2m^2 = 24$$.

The parabola is $$y^{2}=16\sqrt{3}\,x$$.
For $$y^{2}=4ax$$ the slope-form of a tangent is $$y=mx+\frac{a}{m}$$, where $$m\neq 0$$. Here $$4a=16\sqrt{3}\;\Rightarrow\;a=4\sqrt{3}$$, so every tangent of slope $$m$$ is

$$y = mx+\frac{4\sqrt{3}}{m} \qquad -(1)$$

The ellipse is $$2x^{2}+y^{2}=4$$. Rewrite it as $$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$; its centre is the origin.

Take the general line (1): $$y=mx+c$$, where $$c=\dfrac{4\sqrt{3}}{m}$$ from the parabola. Substitute $$y$$ into the ellipse:

$$2x^{2}+(mx+c)^{2}=4$$ $$\Longrightarrow\;(2+m^{2})x^{2}+2mc\,x+(c^{2}-4)=0 \qquad -(2)$$

For (1) to be tangent to the ellipse, the quadratic (2) must have equal roots, i.e. its discriminant is $$0$$:

$$[2mc]^{2}-4(2+m^{2})(c^{2}-4)=0$$ $$\Longrightarrow\;m^{2}c^{2}-(2+m^{2})(c^{2}-4)=0 \qquad -(3)$$

Insert $$c=\dfrac{4\sqrt{3}}{m}$$ into (3):

$$m^{2}\!\left(\frac{16\!\cdot\!3}{m^{2}}\right)-\bigl(2+m^{2}\bigr)\!\left(\frac{48}{m^{2}}-4\right)=0$$ $$\Longrightarrow\;48-(2+m^{2})\!\left(\frac{48-4m^{2}}{m^{2}}\right)=0$$ $$\Longrightarrow\;48m^{2}-\bigl(2+m^{2}\bigr)(48-4m^{2})=0$$ $$\Longrightarrow\;48m^{2}-\bigl(96+40m^{2}-4m^{4}\bigr)=0$$ $$\Longrightarrow\;4m^{4}+8m^{2}-96=0$$ $$\Longrightarrow\;m^{4}+2m^{2}=24 \qquad -(4)$$

Equation (4) is exactly the condition quoted in Statement 2, so Statement 2 is true.

Now solve (4). One obvious root is $$m=2$$ because $$2^{4}+2(2)^{2}=16+8=24$$. For $$m=2$$, the corresponding tangent from (1) is

$$y=2x+\frac{4\sqrt{3}}{2}=2x+2\sqrt{3}$$

This is precisely the line mentioned in Statement 1, and it satisfies both curves, so Statement 1 is also true.

Further, Statement 1 follows directly from Statement 2 because putting the admissible value $$m=2$$ from (4) into the slope-form yields the required tangent. Hence Statement 2 provides the correct explanation for Statement 1.

Option B which is: Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1