The salt $Zn(OH{)}_2$ is involved in the following two equilibria:
$Zn\left(OH{)}_2\right(s)\rightleftharpoons Z{n}^{2+}(aq)+2O{H}^{-}(aq);K_{sp}=1.2\times {10}^{-17}$
$Zn\left(OH{)}_2\right(s)+2O{H}^{-}(aq)\rightleftharpoons [Zn\left(OH{)}_4{]}^{2-}\right(aq);K_c=0.2$
Calculate the pH of the solution at which solubility is minimum.

$a.Zn{\left(OH\right)}_{2\left(s\right)}⟺Z{n}^{+2}+2O{H}^{-}$

$K_{sp}=\left[Z{n}^{+2}\right]{\left[O{H}^{-}\right]}^2$

$b.Zn{\left(OH\right)}_2+2O{H}^{-}⟺{\left[Zn{\left(OH\right)}_2\right]}^{2-}$

$K_f=\frac{{\left[Zn{\left(OH\right)}_4\right]}^{2-}}{\left[O{H}^{2-}\right]}$

By coupling a and b

$2Zn{\left(OH\right)}_{2\left(s\right)}⟺Z{n}_{\left(aq\right)}^{2+}+Zn{\left(OH\right)}_{4\left(aq\right)}^{2-}$

$K_{sp}\times K_f=S\times S$

$S=\sqrt{K_{sp}\times K_f}$

$\sqrt{2\times {10}^{-17}\times 0.2}=1.2\times {10}^{-9}M$

$Thustotalsolubilitywillbe$

$2S=2\times 1.2\times {10}^{-9}M=2.4\times {10}^{-9}M$

The Solubility is minimum when

$\left[Z{n}^{2+}\right]{\left[O{H}^{-}\right]}^2=1.2\times {10}^{-17}$

$1.2\times {10}^{-9}{\left[O{H}^{-}\right]}^2=1.2\times {10}^{-17}$

$\left[O{H}^{-}\right]={10}^{-4}M$

$P^{OH}=4$

$Asweknowthat{p}^H+p^{OH}=14$

$p^H=14-4=10$