Question

The distance between the point $(1,1)$ and the tangent to the curve $y=e^{2x}+x^2$ drawn at the point $x=0$ is

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

Answer 1

Answer: (C)

Putting $x=0$ in $y=e^{2x}+x^2$ we get $y=1$
$\therefore$ The given point is $P(0,1)$
$y=e^{2x}+x^2\frac{dy}{dx}=2e^{2x}+2x$
$\Rightarrow {\left[\frac{dy}{dx}\right]}_P=2$
$\therefore$ Equation of tangent at $P$ to equation (i)
is $y-1=2(x-0)$
$\Rightarrow 2x-y+1=0$
$\therefore$ Required distance $=$ Length of $\perp$ from $(1,1)$ to
equation (ii). $=\frac{2-1+1}{\sqrt{4+1}}=\frac{2}{\sqrt{5}}$.