Question

The distance between the origin and the normal to the curve $ y={{e}^{2x}}+{{x}^{2}} $ at $ x=0 $ is

• A. $ 2 $
• B. $ \frac{2}{\sqrt{3}} $
• C. $ \frac{2}{\sqrt{5}} $
• D. $ \frac{1}{2} $
• E. $ \frac{1}{\sqrt{5}} $

Answer 1

Answer: (C)

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}} $