You are confused #3

It will only work if $|a| > |b|$ .

It won't work if $a > 0$ whereas $b < 0$

Since $\sqrt{a^2}=|a|$ so $a^2=|a|^2$ and since $|a| \in \mathbb{N}$ so it only true if $|a|^2>|b|^2 \implies |a|>|b|$ and the answer is No