A) \[(-\,\infty ,\,1)\]
B) \[(-\,\infty ,\,0)\]
C) \[(1,\,\infty )\]
D) \[(-\,\infty ,\,0)\cup (1,\,\infty )\]
Correct Answer: D
Solution :
We have, \[{{C}_{1}}{{C}_{2}}=5<\]sum of radii But \[{{C}_{1}}{{C}_{2}}=\]difference of radii Thus, the given circles touch each other internally. Hence, number of common tangent is only one. We have, \[1-{{e}^{\frac{1}{x}-1\,}}>0\] \[\Rightarrow \] \[{{e}^{\frac{1}{x}-1}}<1\] \[\Rightarrow \] \[\frac{1}{x}-1<\log 1\] \[\Rightarrow \] \[\frac{1}{x}-1<0\] \[\Rightarrow \] \[\frac{1}{x}<1\] \[\Rightarrow \] \[x\in (-\,\infty ,0)\cup (1,\infty )\]
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