If the tangents on the ellipse $$4x^{2} + y^{2} = 8$$ at the points (1, 2) and (a, b) are perpendicular to each other, then $$a^{2}$$ is equal to:

We begin with the given ellipse

$$4x^{2}+y^{2}=8.$$

For a second-degree curve of the form $$Ax^{2}+By^{2}=C,$$ the equation of the tangent at a point $$(x_{1},y_{1})$$ lying on the curve is obtained by replacing each squared term by the product of the corresponding variables, namely

$$Axx_{1}+Byy_{1}=C.$$

Applying this directly to the ellipse (where $$A=4,\;B=1,\;C=8$$) we see that the tangent at any point $$(x_{1},y_{1})$$ on the ellipse is

$$4xx_{1}+yy_{1}=8.$$

One of the points is explicitly given as $$(1,\,2).$$ Substituting $$x_{1}=1,\;y_{1}=2$$ into the tangent formula gives

$$4x(1)+y(2)=8\;\Longrightarrow\;4x+2y=8.$$

Simplifying by dividing every term by $$2$$ we obtain

$$2x+y=4.$$

To read the slope, we write the line in the form $$y=mx+c:$$

$$y=-2x+4,$$

so the slope of this first tangent is

$$m_{1}=-2.$$

The problem states that the tangent at the unknown point $$(a,\,b)$$ is perpendicular to this one. The product of slopes of two perpendicular lines is $$-1,$$ that is,

$$m_{1}m_{2}=-1.$$

We already have $$m_{1}=-2,$$ so

$$(-2)\,m_{2}=-1\;\Longrightarrow\;m_{2}=\frac12.$$

Now we write the tangent at $$(a,\,b).$$ Using the same tangent formula, we first set up

$$4a\,x+b\,y=8.$$

To identify its slope, we solve for $$y$$:

$$b\,y=8-4a\,x,$$

so

$$y=\frac{8-4a\,x}{b}=-\frac{4a}{b}\,x+\frac{8}{b}.$$

Thus the slope of this second tangent is

$$m_{2}=-\frac{4a}{b}.$$

We already found that $$m_{2}=\dfrac12,$$ therefore

$$-\frac{4a}{b}=\frac12.$$

Cross-multiplying gives

$$2\!\left(-4a\right)=b,$$

or more simply

$$b=-8a.$$

The coordinates $$(a,\,b)$$ must also satisfy the original ellipse equation. Substituting $$b=-8a$$ into $$4a^{2}+b^{2}=8$$ we get

$$4a^{2}+(-8a)^{2}=8.$$

Expanding the squared term,

$$4a^{2}+64a^{2}=8,$$

which combines to

$$68a^{2}=8.$$

Solving for $$a^{2}$$ yields

$$a^{2}=\frac{8}{68}=\frac{4}{34}=\frac{2}{17}.$$

Hence, the correct answer is Option A.