jmc

From the given information we know that $|a||b|=|ab|=2\sqrt{26}$. We can also write $|ab|=|t-2i|=\sqrt{t^2+4}$. Setting these equal, we have $$\sqrt{t^2+4}=2\sqrt{26}\Rightarrow t^2+4=104.$$The positive answer is $t=\boxed{10}$.