See [this earlier MO question][1]. The union of n disks (represented in terms of its boundary arcs) can be constructed in $O(n\log n)$ time. So construct the union of the $n$ disks you are given, and separately construct the union of the $n+1$ disks including the one you want covered. The given disk is covered if and only if the two unions are the same.


  [1]: http://mathoverflow.net/questions/24859/finding-the-union-of-n-random-circles-arbitrarily-or-conspiratorially-placed-on