The point of intersection of the tangent to the parabola $y^2=4x$ which also touches $x^2+y^2=\frac{1}{2}$ is

Equation of the tangent to the parabola is $y=mx+\frac{1}{m}$
Equation of the tangent to the circle is $y=mx\pm \frac{1}{\sqrt{2}}\sqrt{1+m^2}$
For common tangent
$\frac{1}{m^2}=\frac{1}{2}\left(1+m^2\right)\Rightarrow m^4+m^2-2=0\Rightarrow m^2=1\Rightarrow m=\pm 1$
$\Rightarrow$ Equation of the tangents are $y=x+1$ and $y=-x-1$
$\Rightarrow$ Point of intersection of these tangents is $\left(-1,0\right)$