Choose the incorrect statement about the two circles whose equations are given below:
$$x^2 + y^2 - 10x - 10y + 41 = 0$$ and $$x^2 + y^2 - 16x - 10y + 80 = 0$$

The two circles are $$x^2 + y^2 - 10x - 10y + 41 = 0$$ and $$x^2 + y^2 - 16x - 10y + 80 = 0$$.

For the first circle, completing the square: $$(x-5)^2 + (y-5)^2 = 25 + 25 - 41 = 9$$. So centre $$C_1 = (5, 5)$$ and radius $$r_1 = 3$$.

For the second circle: $$(x-8)^2 + (y-5)^2 = 64 + 25 - 80 = 9$$. So centre $$C_2 = (8, 5)$$ and radius $$r_2 = 3$$.

The distance between centres is $$d = \sqrt{(8-5)^2 + (5-5)^2} = 3$$.

Now let us check each statement:

Option A: "Distance between centres is the average of radii." The average of $$r_1$$ and $$r_2$$ is $$\frac{3+3}{2} = 3 = d$$. This is TRUE.

Option B: "Both circles' centres lie inside the region of one another." For $$C_2 = (8,5)$$ to be inside circle 1, we need $$(8-5)^2 + (5-5)^2 = 9 < r_1^2 = 9$$. Since $$9$$ is not strictly less than $$9$$, $$C_2$$ lies ON circle 1, not inside it. Similarly $$C_1$$ lies ON circle 2. So this statement is INCORRECT.

Option C: "Both circles pass through the centre of each other." Substituting $$C_2 = (8,5)$$ into circle 1: $$(8-5)^2 + (5-5)^2 = 9 = r_1^2$$. Yes, $$C_2$$ lies on circle 1. Similarly $$(5-8)^2 + (5-5)^2 = 9 = r_2^2$$, so $$C_1$$ lies on circle 2. This is TRUE.

Option D: "Circles have two intersection points." Since $$|r_1 - r_2| < d < r_1 + r_2$$ becomes $$0 < 3 < 6$$, which is true, the circles intersect at two points. This is TRUE.

The incorrect statement is Option B, since the centres lie ON the other circle, not inside it.