Question

The perpendicular 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. $\frac{3}{\sqrt{5}}$
• C. $\frac{2}{\sqrt{5}}$
• D. $\frac{4}{\sqrt{5}}$

Answer 1

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

The equation of given curve is $y=e^{2x}+x^2$........(i)

At $x=0\Rightarrow y=1$

$\therefore$ the given point is P(0, 1)

$\frac{dy}{dx}=2e^{2x}+2x$

Slope of tangent at (0,1) is $(\frac{dy}{dx}{)}_{(0,1)}=2$

Equation of tangent at P(0, 1) to $e{q}^n\left(i\right)$ is

$y-1=2(x-0)$

$\Rightarrow 2x-y+1=0$

$\therefore$ Required distance $=$ length of perpendicular from (1, 1) to the line $2x-y+1=0$

$=|\frac{2-1+1}{\sqrt{4+1}}|=\frac{2}{\sqrt{5}}$