A) \[\frac{1}{\sqrt{5}}\]
B) \[\frac{2}{\sqrt{5}}\]
C) \[-\frac{1}{\sqrt{5}}\]
D) \[\frac{2}{\sqrt{3}}\]
Correct Answer: A
Solution :
The equation of given curve is \[y={{e}^{2x}}+{{x}^{2}}\] ...(i) At \[x=0,y=e{}^\circ +0=1\] On differentiating Eq. (i) w.r.t. x, we get \[\frac{dy}{dx}=2{{e}^{2x}}+2x\] \[\Rightarrow \] \[{{\left( \frac{dy}{dx} \right)}_{(0,1)}}=2{{e}^{0}}+0=2\] The equation of the tangent at point (0,1) is \[y-1=2(x-0)\] \[\Rightarrow \] \[2x-y+1=0\] \[\therefore \]Required distance = length of perpendicular from point (0,0) to 2x - y + 1 = 0\[=\frac{|0-0+1|}{\sqrt{4+1}}=\frac{1}{\sqrt{5}}\]
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