The tangent drawn at the point$(0, 1)$ on the curve $y=e^{2x}$ meets x-axis at the point: 

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

$\Rightarrow {\left(\frac{dy}{dx}\right)}_{(0,1)}=2$

Equation of tangent is, $y - 1 = 2(x - 0) \Rightarrow y = 2x + 1$

This tangent meets x-axis,$:. y = 0$

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

Co-ordinates of the point $\left(-\frac{1}{2},0\right)$