If (3, -6) is the mid-point of the line segment joining (0, 0) and (x, y), then the point (x, y) is:

1. (-3, 6)

2. (6, -6)

3. (6, -12)

4. $\Big(\dfrac{3}{2}, -3\Big)$

Given, (3, -6) is the mid-point of the line segment joining (0, 0) and (x, y).

$\therefore (3, -6) = \Big(\dfrac{0 + x}{2}, \dfrac{0 + y}{2}\Big) \\[1em] \Rightarrow (3, -6) = \Big(\dfrac{x}{2}, \dfrac{y}{2}\Big) \\[1em] \Rightarrow 3 = \dfrac{x}{2} \text{ and } -6 = \dfrac{y}{2} \\[1em] \Rightarrow x = 6 \text{ and } y = -12$

(x, y) = (6, -12).

Hence, Option 3 is the correct option.