Given the relation $R=\{(a,b):a=b-2, \; b>6 \}$ in a set N, we can arrive at the right option by substituting them in the relation to see which one holds true:

$(2,4) \in R \Rightarrow 2 = 4-2 = 2$, which is ok, but since $b>6$, this cannot be the right answer.

$(3,8) \in R \Rightarrow a = 3$ and $b-2 =8 -2 = 6$, which are not equal. So this is not the right answer either.

$(6,8) \in R \Rightarrow a = 6$ and $b-2=8 -2 =6$, which are equal. Also since $b=8 \gt 6$, this is the right answer.

$(8,7) \in R \Rightarrow a = 8$ and $b-2 =7 -2 = 5$, which are not equal. So this is not the right answer either.