$\text{ If }|z-1|=|z+1|=|z-2i|,\text{ then value of }\left|z\right|\text{ is }\frac{a}{b}\text{, hence, a+b=}$ $Here ( a,b) are coprime numbers$

$z\text{ is centre of the circle passing through }1+0i,-1+0i\text{ and }0+2i\text{ . Clearly centre lies on the }y\text{ - }$axis

$\text{ If }z=0+ai\text{ the centre, then }$

$\sqrt{1+a^2}=|a+2|$

$\Rightarrow 1+a^2=a^2+4a+4$

$\Rightarrow a=\frac{3}{4}$

$\therefore \left|z\right|=\frac{3}{4}$