Answer
See the answer below.
Work Step by Step
$B.O.=N.\ Shared\ pairs\div N.\ links$
$NO_2^+$: $4/2=2$
$NO_2^-$: $3/2=1.5$
$NO_3^-$: $4/3=1.33$
Since bond length decreases with bond order, the NO bonds will be the longest in $NO_3^-$ and the shortest in $NO_2^+$.
