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 $y=e^{2x}+x^2$
$\therefore$ $x=0$
$\Rightarrow$ $y=1$
$\Rightarrow$ $\frac{dy}{dx}=2e^{2x}+2x$
$=2(e^{2x}+x)$
$\therefore$ ${\left(\frac{dy}{dx}\right)}_{(0,1)}=2$
$\therefore$ Equation of normal at (0, 1) is $y-1=-\frac{1}{2}(x-0)$
$\Rightarrow$ $2y+x=2$
$\therefore$ Required distance
$=\left|\frac{-2}{\sqrt{2^2+1^2}}\right|$
$=\left|-\frac{2}{\sqrt{5}}\right|$
$=\frac{2}{\sqrt{5}}$