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 $P\left(0,1\right)$
$y=e^{2x}+x^2\cdots \left(1\right)$
$\Rightarrow \frac{dy}{dx}=2e^{2x}+2x$
$\Rightarrow {\left[\frac{dy}{dx}\right]}_P=2$
$\therefore$ Equation of tangent at $P$ to $\left(1\right)$ is
$y-1=2\left(x-0\right)$
$\Rightarrow 2x-y+1=0\cdots \left(2\right)$
$\therefore$ Required distance = Length of $\perp$ from $\left(1,1\right)$ to $\left(2\right)$
$=\frac{2-1+1}{\sqrt{4+1}}=\frac{2}{\sqrt{5}}$.