Let $$z_1 = 2 + 3i$$ and $$z_2 = 3 + 4i$$. The set $$S = \{z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2\}$$ represents a

We are given $$z_1 = 2 + 3i$$ and $$z_2 = 3 + 4i$$ and need to determine the set $$S = \{z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2\}$$.

To begin, compute $$|z_1 - z_2|^2$$. We have $$z_1 - z_2 = (2 + 3i) - (3 + 4i) = -1 - i$$ and therefore $$|z_1 - z_2|^2 = (-1)^2 + (-1)^2 = 1 + 1 = 2$$.

Next, let $$z = x + iy$$ and expand the squared distances. Using $$|z - a|^2 = (x - \operatorname{Re}(a))^2 + (y - \operatorname{Im}(a))^2$$, it follows that $$|z - z_1|^2 = (x - 2)^2 + (y - 3)^2 = x^2 - 4x + 4 + y^2 - 6y + 9$$ and $$|z - z_2|^2 = (x - 3)^2 + (y - 4)^2 = x^2 - 6x + 9 + y^2 - 8y + 16$$.

Subtracting these expressions gives
$$(x^2 - 4x + 4 + y^2 - 6y + 9) - (x^2 - 6x + 9 + y^2 - 8y + 16)$$
$$= -4x + 6x + 4 - 9 - 6y + 8y + 9 - 16$$
$$= 2x + 2y - 12$$.

Setting this equal to $$|z_1 - z_2|^2 = 2$$ leads to $$2x + 2y - 12 = 2$$, so $$2x + 2y = 14$$ and $$x + y = 7$$. This is the equation of a straight line.

The x-intercept occurs when $$y = 0$$, giving $$x = 7$$, and the y-intercept occurs when $$x = 0$$, giving $$y = 7$$. The sum of these intercepts is $$7 + 7 = 14$$.

Therefore the correct answer is Option A.