Four distinct points $$(2k, 3k), (1, 0), (0, 1)$$ and $$(0, 0)$$ lie on a circle for $$k$$ equal to :

The circle passes through $$(0, 0)$$, $$(1, 0)$$, $$(0, 1)$$, and $$(2k, 3k)$$.

The general equation of a circle through the origin: $$x^2 + y^2 + Dx + Ey = 0$$.

Using $$(1, 0)$$: $$1 + D = 0 \Rightarrow D = -1$$.

Using $$(0, 1)$$: $$1 + E = 0 \Rightarrow E = -1$$.

Circle: $$x^2 + y^2 - x - y = 0$$.

Using $$(2k, 3k)$$:

$$4k^2 + 9k^2 - 2k - 3k = 0$$

$$13k^2 - 5k = 0$$

$$k(13k - 5) = 0$$

Since the four points are distinct and $$k \neq 0$$ (otherwise $$(2k, 3k) = (0,0)$$):

$$k = \frac{5}{13}$$

The answer is $$\frac{5}{13}$$, which corresponds to Option (3).