A) \[\frac{1}{e}\]
B) \[\frac{1}{2e}\]
C) \[\frac{1}{3e}\]
D) \[\frac{1}{4e}\]
Correct Answer: C
Solution :
[c] Curves touch each other. Solving equations of curves, we get \[\frac{{{\log }_{e}}x}{x}=\lambda {{x}^{2}}\] \[\Rightarrow \,\,\,\,\,{{\log }_{e}}x=\lambda {{x}^{3}}\] ?..(1) Comparing their derivatives, we get \[\frac{1-{{\log }_{e}}x}{{{x}^{2}}}=2\lambda x\] \[\Rightarrow \,\,\,1-\lambda {{x}^{3}}=2\lambda {{x}^{3}}\] \[\Rightarrow \,\,\,3\lambda {{x}^{3}}=1\] \[\Rightarrow \,\,\,x={{\left( \frac{1}{3\lambda } \right)}^{\frac{1}{3}}}={{(3\lambda )}^{\frac{-1}{3}}}\] Putting the value of x into (1), we get \[\frac{-1}{3}{{\log }_{e}}(3\lambda )=\lambda .\frac{1}{3\lambda }\] \[\Rightarrow \,\,\,\,{{\log }_{e}}(3\lambda )=1\] \[\Rightarrow \,\,\,\,\lambda =\frac{1}{3e}\]
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