Question

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

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

Answer 1

The correct option is B $\frac{2}{\sqrt{5}}$

$y^{\prime }=2e^{2x}+2x$
$y=1$ at $x=0$
$y^{\prime }=2$

$m$ of normal $=\frac{-1}{2}$

$(y-1)=\frac{-1}{2}(x-0)$

$x+2y-2=0$

distance from origin

$d=\left|\frac{x_1+2y_1-2}{\sqrt{1^2+2^2}}\right|\left(x_1{y}_1\right)=(0,0)$

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