If $$z = x + iy$$ satisfies $$z - 2 = 0$$ and $$z-i - z+5i = 0$$, then

We need to find the relationship between $$x$$ and $$y$$ when $$z = x + iy$$ satisfies $$|z| - 2 = 0$$ and $$|z - i| - |z + 5i| = 0$$.

$$|z| = 2 \Rightarrow x^2 + y^2 = 4$$

$$|z - i| = |z + 5i|$$

$$|x + i(y-1)| = |x + i(y+5)|$$

$$x^2 + (y-1)^2 = x^2 + (y+5)^2$$

$$y^2 - 2y + 1 = y^2 + 10y + 25$$

$$-2y + 1 = 10y + 25$$

$$-12y = 24$$

$$y = -2$$

$$x^2 + (-2)^2 = 4$$

$$x^2 = 0$$

$$x = 0$$

Option A: $$x + 2y - 4 = 0 + (-4) - 4 = -8 \neq 0$$

Option B: $$x^2 + y - 4 = 0 + (-2) - 4 = -6 \neq 0$$

Option C: $$x + 2y + 4 = 0 + (-4) + 4 = 0$$ $$\checkmark$$

Option D: $$x^2 - y + 3 = 0 - (-2) + 3 = 5 \neq 0$$

Therefore, the correct answer is Option C: $$x + 2y + 4 = 0$$.