The locus of the mid points of the chords of the hyperbola $$x^2 - y^2 = 4$$, which touch the parabola $$y^2 = 8x$$, is:

Let the mid-point of a variable chord of the hyperbola $$x^{2}-y^{2}=4$$ be $$(h,k)$$. For any conic whose equation is written as $$S(x,y)=0,$$ the equation of the chord whose mid-point is $$(h,k)$$ is obtained from the formula $$T=S(h,k).$$

We first write the hyperbola in the form $$S(x,y)=0$$:

$$S(x,y)=x^{2}-y^{2}-4=0.$$

Replacing $$x^{2}$$ by $$x\,h$$ and $$y^{2}$$ by $$y\,k$$ we get

$$T=xh-\;y k-4.$$

Next we evaluate $$S(h,k)$$:

$$S(h,k)=h^{2}-k^{2}-4.$$

The required chord therefore satisfies

$$T=S(h,k)\;\;\Longrightarrow\;\;xh-yk-4=h^{2}-k^{2}-4.$$

Simplifying, the chord is

$$h\,x-k\,y=h^{2}-k^{2}.\qquad(1)$$

This straight line is given to be tangent to the parabola $$y^{2}=8x.$$ For the standard parabola $$y^{2}=4ax$$ (here $$4a=8\;\Rightarrow\;a=2$$) the slope form of a tangent is

$$y=mx+\frac{a}{m}=mx+\frac{2}{m}.$$(2)

We now bring equation (1) to the same form so that we may compare the coefficients. Solving (1) for $$y$$ we have

$$k\,y=h\,x-(h^{2}-k^{2})$$

$$\Longrightarrow\;y=\frac{h}{k}\,x-\frac{h^{2}-k^{2}}{k}.$$(3)

From (3) the slope is

$$m=\frac{h}{k},$$

and the $$x$$-intercept term is

$$c=-\frac{h^{2}-k^{2}}{k}.$$

Because (3) must coincide with the general tangent (2), both the slopes and the constant terms must match. We already have the equality of slopes through $$m=\dfrac{h}{k}.$$ Equating the constant terms,

$$-\frac{h^{2}-k^{2}}{k}=\frac{2}{m}=\frac{2k}{h}.$$

Clearing denominators step by step:

$$-\bigl(h^{2}-k^{2}\bigr)=\frac{2k^{2}}{h}$$

$$\Longrightarrow\;-h\bigl(h^{2}-k^{2}\bigr)=2k^{2}$$

$$\Longrightarrow\;-h^{3}+h\,k^{2}=2k^{2}$$

$$\Longrightarrow\;h^{3}-h\,k^{2}+2k^{2}=0.$$

Collecting $$k^{2}$$ as a common factor in the last two terms gives

$$h^{3}=k^{2}(h-2).$$

Finally, replacing the fixed parameters $$h$$ and $$k$$ by the general coordinates $$x$$ and $$y$$ of the sought locus, we obtain

$$y^{2}(x-2)=x^{3}.$$

This is exactly the equation listed in Option A.

Hence, the correct answer is Option A.