If one end of a focal chord $$AB$$ of the parabola $$y^2 = 8x$$ is at $$A\left(\frac{1}{2}, -2\right)$$, then the equation of the tangent to it at $$B$$ is:

We start with the parabola $$y^{2}=8x$$.

First we bring it to the standard form $$y^{2}=4ax$$ so that we can use all known formulae directly.

Comparing $$y^{2}=8x$$ with $$y^{2}=4ax$$, we obtain $$4a = 8 \;\Longrightarrow\; a = 2.$$

For the parabola $$y^{2}=4ax$$, the focus is always at $$(a,0)$$. Hence here the focus is $$(2,0).$$

Any point on this parabola can be expressed parametrically. With the parameter $$t$$, the coordinates are

$$\bigl(at^{2},\,2at\bigr) = \bigl(2t^{2},\,4t\bigr).$$

Let the parameters corresponding to the two ends of the focal chord $$AB$$ be $$t_{1}$$ and $$t_{2}$$. A standard property of a focal chord of a parabola $$y^{2}=4ax$$ is

$$t_{1}\,t_{2} = -1.$$

We are given the coordinates of the point $$A\Bigl(\dfrac12,\,-2\Bigr)$$. We now identify $$t_{1}$$ by equating these coordinates to $$(2t_{1}^{2},\,4t_{1})$$.

From the $$x$$-coordinate:

$$2t_{1}^{2} = \dfrac12 \;\Longrightarrow\; t_{1}^{2} = \dfrac14 \;\Longrightarrow\; t_{1} = \pm \dfrac12.$$

From the $$y$$-coordinate:

$$4t_{1} = -2 \;\Longrightarrow\; t_{1} = -\dfrac12.$$

Both equations agree only for $$t_{1} = -\dfrac12.$$

Using the focal-chord relation $$t_{1} t_{2} = -1$$, we now find $$t_{2}$$:

$$\left(-\dfrac12\right) t_{2} = -1 \;\Longrightarrow\; t_{2} = 2.$$

So the coordinates of the second end $$B$$ are obtained by substituting $$t_{2}=2$$ in $$(2t^{2},4t)$$:

$$x_{B} = 2\,(2)^{2} = 2 \times 4 = 8,$$

$$y_{B} = 4\,(2) = 8.$$

Hence $$B(8,\,8).$$

For a general point $$(at^{2},\,2at)$$ on $$y^{2}=4ax$$, the tangent has the well-known form

$$t\,y = x + a\,t^{2}.$$

We now substitute $$a = 2$$ and the specific parameter $$t = t_{2} = 2$$:

$$2\,y = x + 2\,(2)^{2}.$$

Step by step simplification:

$$(2)^{2} = 4,$$

$$2 \times 4 = 8,$$

so the equation becomes

$$2y = x + 8.$$

Bring every term to the left to obtain the standard linear form:

$$x - 2y + 8 = 0.$$

Thus the required tangent at $$B$$ is

$$x - 2y + 8 = 0.$$

Hence, the correct answer is Option B.