The set $$\{z\in\mathbb{C}:(2+6i)z+(9-i)\overline{z}+10=0\}$$ on the Argand plane represents:

Substitute $$z = x + i y$$ and $$\bar{z} = x - i y$$ into the given equation.

Expanding the first term:

$$(2 + 6i)(x + i y) = 2x + 2i y + 6i x + 6i^2 y = 2x - 6y + 6i x + 2i y$$

• Real part: $$2x - 6y$$
  
• Imaginary part: $$6x + 2y$$
  

Expanding the second term:

$$(9 - i)(x - i y) = 9x - 9i y - i x + i^2 y = 9x - y - 9i x - i y$$

• Real part: $$9x - y$$
  
• Imaginary part: $$-9x - y$$
  

Adding both real parts and the constant $$10$$ to equal zero:

$$2x - 6y + 9x - y + 10 = 0$$

$$11x - 7y + 10 = 0$$

Adding both imaginary parts to equal zero:

$$6x + 2y - 9x - y = 0$$

$$-3x + y = 0$$

From the imaginary equation, we get $$y = 3x$$.

Substituting this into the real equation:

$$11x - 7(3x) + 10 = 0$$

$$11x - 21x + 10 = 0$$

$$-10x + 10 = 0$$

$$x = 1$$

Using $$y = 3x$$, we get $$y = 3$$.

The unique solution is $$z = 1 + 3i$$, which represents a single point on the Argand plane.

The correct option is D.