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(B) For the first circle $x^{2}+y^{2}-10x-10y+41=0$:Centre $C_{1} = (5, 5)$,Radius $r_{1} = \sqrt{5^{2}+5^{2}-41} = \sqrt{25+25-41} = \sqrt{9} = 3$.Fo

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Both circles' centres lie inside the region of one another.

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(B) For the first circle $x^{2}+y^{2}-10x-10y+41=0$:Centre $C_{1} = (5, 5)$,Radius $r_{1} = \sqrt{5^{2}+5^{2}-41} = \sqrt{25+25-41} = \sqrt{9} = 3$.For the second circle $x^{2}+y^{2}-16x-10y+80=0$:Centre $C_{2} = (8, 5)$,Radius $r_{2} = \sqrt{8^{2}+5^{2}-80} = \sqrt{64+25-80} = \sqrt{9} = 3$.The distance between the centres is $d = \sqrt{(8-5)^{2}+(5-5)^{2}} = \sqrt{3^{2}+0^{2}} = 3$.Since $d = r_{1} = r_{2} = 3$,the distance between the centres is equal to the radius of each circle.This implies that each circle passes through the centre of the other circle.Statement $A$ is incorrect because the distance between the centres is $3$,while the average of the radii is $(3+3)/2 = 3$. Wait,$3=3$,so $A$ is actually correct.Let's re-evaluate: $C_{1}(5,5), C_{2}(8,5), r_{1}=3, r_{2}=3, d=3$.$A$: $d=3$,average radius $= 3$. Correct.$B$: $C_{2}$ is at distance $3$ from $C_{1}$,so $C_{2}$ is on the circumference of circle $1$,not inside. Thus,$B$ is incorrect.$C$: $d=r_{1}=r_{2}$,so each passes through the other's centre. Correct.$D$: Since $|r_{1}-r_{2}| Therefore,the incorrect statement is $B$.

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

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