Chemistry - Ionic Equilibrium Question with Solution | TestHub

$\begin{array}{rl}& {\mathrm{HX}}_{(\mathrm{aq})}\rightleftharpoons {\mathrm{H}}_{(\mathrm{aq})}^{+}+{\mathrm{X}}_{(\mathrm{aq})}^{-}{\mathrm{K}}_{\mathrm{a}}=4\times 10^{-10}\\ & 0.01(1-\alpha )0.01\alpha 0.01\alpha \text{ Not justified }\\ & \Rightarrow 0.01\alpha =10^{-5}\Rightarrow \alpha =10^{-3}\\ & {\mathrm{K}}_{\mathrm{a}}=0.01{\alpha }^2=10^{-8}\end{array}$
On dilution let final concentration of $\mathrm{HX}=\mathrm{cM}$
$\begin{array}{rl}& {\mathrm{Hx}}_{(\mathrm{an})}\rightleftharpoons {\mathrm{H}}_{(\mathrm{aq})}^{+}+{\mathrm{X}}_{(\mathrm{an})}^{-}\\ & \mathrm{C}(1-\alpha )\mathrm{C}\alpha \mathrm{C}\alpha \\ & \Rightarrow \mathrm{C}\alpha =10^{-6}...(1)\\ & \frac{\mathrm{C}{\alpha }^2}{1-\alpha }={\mathrm{K}}_{\mathrm{a}}=10^{-8}...(2)\\ & \Rightarrow \frac{10^{-6}\alpha }{1-\alpha }=10^{-8}\end{array}$
Data given is inconsistent \& contradictory. This should be bonus.