A circle has its centre at (4, 4). If one end of a diameter is (4, 0), then the coordinates of the other end are:

1. (0, 4)

2. (4, 8)

3. (4, -8)

4. (-4, -8)

Let one end of the diameter be A(4, 0) and the other end be B(x, y). Given that the centre of the circle is (4, 4), which is the mid-point of AB.

By mid-point formula,

(x, y) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get :

$\Rightarrow (4, 4) = \Big(\dfrac{4 + x}{2}, \dfrac{0 + y}{2}\Big) \\[1em] \Rightarrow 4 = \dfrac{4 + x}{2} \text{ and } 4 = \dfrac{y}{2} \\[1em] \Rightarrow x + 4 = 8 \text{ and } y = 8 \\[1em] \Rightarrow x = 8 - 4 = 4 \text{ and } y = 8.$

The coordinates of the other end of the diameter are (4, 8).

Hence, Option 2 is the correct option.