A) \[{{\tan }^{-1}}\left( \frac{\log 3-\log 5}{1+\log 3\,\,\log 5} \right)\]
B) \[{{\tan }^{-1}}\left( \frac{\log 3+\log 5}{1-\log 3\,\,\log 5} \right)\]
C) \[{{\tan }^{-1}}\left( \frac{\log 3+\log 5}{1+\log 3\,\,\log 5} \right)\]
D) \[{{\tan }^{-1}}\left( \frac{\log 3-\log 5}{1-\log 3\,\,\log 5} \right)\]
Correct Answer: A
Solution :
Given curves \[y={{3}^{x}}\] ...(i) and \[y={{5}^{x}}\] ... (ii) Intersect at the point (0, 1). Now, differentiating Eqs. (i) and (ii) w.r.t. x, we get \[\frac{dy}{dx}={{3}^{x}}\log 3\]and \[\frac{dy}{dx}={{5}^{x}}\log 5\] \[\Rightarrow \]\[{{\left( \frac{dy}{dx} \right)}_{(0,\,1)}}=\log \,3\]and \[{{\left( \frac{dy}{dx} \right)}_{(0,\,1)}}=\log \,5\] \[\Rightarrow \] \[{{m}_{1}}=\log 3\]and \[{{m}_{2}}=\log 5\] Angle between these curves is given by \[\tan \theta =\frac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}}\] \[\Rightarrow \] \[\tan \theta =\frac{\log 3-\log 5}{1+\log 3\cdot \log 5}\] \[\Rightarrow \] \[\theta ={{\tan }^{-1}}\left( \frac{\log 3-\log 5}{1+\log 3\,\,\log 5} \right)\]
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