A) \[r\,\,>\,\,11\]
B) \[r\,\,=\,\,11\]
C) \[1\,\,<\,\,r\,\,<\,\,11\]
D) \[0\,\,<\,\,r\,\,<\,\,1\]
Correct Answer: C
Solution :
Circles are \[{{x}^{2}}+{{y}^{2}}-16x-20y+164={{r}^{2}}\] \[\Rightarrow \,\,\,\,{{c}_{1}}\left( 8,\text{ }10 \right)\] and \[{{\left( x-4 \right)}^{2}}+{{\left( y-7 \right)}^{2}}=36\] they intersect at two distinct points \[\left| {{r}_{1}}-{{r}_{2}} \right|<{{c}_{1}}{{c}_{2}}<{{r}_{1}}+{{r}_{2}}\left\{ {{c}_{1}}{{c}_{2}}=\sqrt{16+9}=5 \right\}\] Now \[\left| r-6 \right|<5<r+6\] \[\left| r-6 \right|<5~~~~~~~~~~~5<r+6\] \[\Rightarrow \,\,-5<r-6<5~~~~~~-1\,\,<\,\,r~~...\text{ }\left( ii \right)\] \[\Rightarrow \,\,1<r<11~~~~~~\,\,\,\,~~...\text{ }\left( i \right)\] from (i) and (ii) \[r\text{ }\in \text{ }\left( 1,\text{ }11 \right)\]
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