The equation of the circle passing through the foci of the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$, and having centre at (0, 3) is

We have the given ellipse $$\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1.$$

For an ellipse of the form $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ with its major axis along the x-axis, the semi-major axis is $$a$$, the semi-minor axis is $$b$$, and the distance of each focus from the centre (denoted by $$c$$) satisfies the relation $$c^2 = a^2 - b^2.$$

Comparing the given equation with the standard form, we identify

$$a^2 = 16 \;\;\Rightarrow\;\; a = 4,$$

$$b^2 = 9 \;\;\Rightarrow\;\; b = 3.$$

Applying the formula $$c^2 = a^2 - b^2,$$ we obtain

$$c^2 = 16 - 9 = 7 \;\;\Longrightarrow\;\; c = \sqrt{7}.$$

Hence the coordinates of the two foci are

$$\bigl(\sqrt{7},\,0\bigr) \quad\text{and}\quad \bigl(-\sqrt{7},\,0\bigr).$$

Now we need a circle that passes through these two foci and whose centre is fixed at the point $$(0,\,3).$$ For any circle, every point on the circle is at the same distance (the radius) from the centre. Therefore we can find the radius by computing the distance from the centre $$(0,\,3)$$ to either focus, say $$(\sqrt{7},\,0).$$

The distance formula between two points $$(x_1,\,y_1)$$ and $$(x_2,\,y_2)$$ is stated as

$$\text{Distance} \;=\; \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$

Substituting $$(x_1,\,y_1) = (0,\,3)$$ and $$(x_2,\,y_2) = (\sqrt{7},\,0),$$ we get

$$r = \sqrt{(\sqrt{7} - 0)^2 + (0 - 3)^2} = \sqrt{(\sqrt{7})^2 + (-3)^2} = \sqrt{7 + 9} = \sqrt{16} = 4.$$

So the required circle has centre $$(0,\,3)$$ and radius $$4.$$ The standard equation of a circle with centre $$(h,\,k)$$ and radius $$r$$ is

$$\bigl(x - h\bigr)^2 + \bigl(y - k\bigr)^2 = r^2.$$

Putting $$h = 0,$$ $$k = 3,$$ and $$r = 4,$$ we obtain

$$\bigl(x - 0\bigr)^2 + \bigl(y - 3\bigr)^2 = 4^2,$$

$$x^2 + (y - 3)^2 = 16.$$

Expanding the square in the y-term,

$$x^2 + \bigl(y^2 - 6y + 9\bigr) = 16.$$

Combining like terms,

$$x^2 + y^2 - 6y + 9 - 16 = 0,$$

$$x^2 + y^2 - 6y - 7 = 0.$$

This matches Option C.

Hence, the correct answer is Option C.