The distance between the origin and the tangent to the curve y=e^2 x+…

Putting x = 0 in the given curve, we obtain y = 1

So, the given point is (0, 1)

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

The equation of the tangent at (0, 1) is

$y-1=2(x-0) \Rightarrow 2 x-y+1=0$          ..........(i)

⇒ Required distance = Length of the ⊥ from (0, 0) on (i)

⇒ Required distance = $\frac{1}{\sqrt{5}}$