If the common tangent to the parabolas $y^2=4xand{x}^2=4y$ also touches the circle $x^2+y^2=c^2$ then $c=$

Tangent to parabola $y^2=4x$ is $y=mx+\frac{1}{m}$

Tangent to parabola $x^2=4y$ is $y=mx-m^2$

$\frac{1}{m}=-m^2\Rightarrow m=-1$

Equation of common tangent is $x+y+1=0$ is tangent to $x^2+y^2=c^2\Rightarrow C=\frac{1}{\sqrt{2}}$