A straight line is normal to both the parabolas $y^2=x\text{ and }{x}^2=y\text{. }$ The distance of the origin from it is  

The equation of the common normal to $y^2=4ax\text{ and }{x}^2=4ay\text{ is }x+y=3a$

Here , $a=\frac{1}{4}$ . Its equation is $x+y=\frac{3}{4}$

The distance of (0, 0) from it is $\frac{3}{4\sqrt{2}}$