A) \[2\]
B) \[\frac{2}{\sqrt{3}}\]
C) \[\frac{2}{\sqrt{5}}\]
D) \[\frac{1}{2}\]
E) \[\frac{1}{\sqrt{5}}\]
Correct Answer: C
Solution :
Given curve is \[y={{e}^{2x}}+{{x}^{2}}\] \[\therefore \] \[x=0\] \[\Rightarrow \] \[y=1\] \[\Rightarrow \] \[\frac{dy}{dx}=2{{e}^{2x}}+2x\] \[=2({{e}^{2x}}+x)\] \[\therefore \] \[{{\left( \frac{dy}{dx} \right)}_{(0,1)}}=2\] \[\therefore \]Equation of normal at (0, 1) is \[y-1=-\frac{1}{2}(x-0)\] \[\Rightarrow \] \[2y+x=2\] \[\therefore \]Required distance \[=\left| \frac{-2}{\sqrt{{{2}^{2}}+{{1}^{2}}}} \right|\] \[=\left| -\frac{2}{\sqrt{5}} \right|\] \[=\frac{2}{\sqrt{5}}\]
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