Question

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

• A. $\sqrt{3}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{2}{\sqrt{5}}$
• D. None of these

Answer 1

The correct option is C

$\frac{2}{\sqrt{5}}$

Explanation for the correct option:

Find the distance between the origin and the normal of the curve:

Given $y=e^{2x}+x^2$ at $x=0$

$\mathrm{At}x=0\Rightarrow y=1$

Differentiate given function with respect to '$x$'

$$\begin{gathered} \frac{dy}{dx}=2e^{2x}+2x \\ {\left(\frac{dy}{dx}\right)}_{(0,1)}=2 \end{gathered}$$

Slope of the normal is given by

$$\begin{gathered} \Rightarrow m=\frac{-1}{{\left(\frac{dy}{dx}\right)}_{(0,1)}} \\ \Rightarrow m=-\frac{1}{2} \end{gathered}$$

Equation of normal at $(0,1)$ is

$$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-1=\frac{-1}{2}(x-0) \\ \Rightarrow 2y+x=2 \end{gathered}$$

The required distance between the origin and the normal to the curve is:

$$\begin{gathered} d=\begin{array}{rcl}& & \left|\frac{2y+x-2}{\sqrt{a^2+b^2}}\right|\text{at}(0,0)\end{array} \\ =\left|\frac{2\left(0\right)+0-2}{\sqrt{2^2+1^2}}\right| \\ =\left|\frac{-2}{\sqrt{5}}\right| \\ =\frac{2}{\sqrt{5}} \end{gathered}$$

Hence, option (C) is the correct answer.