Question

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

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

Answer 1

The correct option is A $\frac{1}{\sqrt{5}}$

Putting $x=0$ in the given curve, we obtain $y=1.$
So, the given point is $(0,1).$

Now, $y=e^{2x}+x^2$
$\Rightarrow \frac{dy}{dx}=2e^{2x}+2x$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{0,1}=2$

The equation of the tangent at $(0,1)$ is
$y-1=2(x-0)$
$\Rightarrow 2x-y+1=0...\left(1\right)$

Required distance is the length of perpendicular from $(0,0)$ on the line $\left(1\right)$
Required distance $=\frac{1}{\sqrt{5}}$