To determine which line the midpoint $R$ of $P$ and $Q$ lies on, follow these steps:

1. Find the midpoint $R$ of $P$ and $Q$:

The midpoint $R$ of a line segment with endpoints $P(x_1,y_1)$ and $Q(x_2,y_2)$ is given by:
$$R=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

For points $P(3,5)$ and $Q(-1,9)$:

$$R=(\frac{3+(-1)}{2},\frac{5+9}{2})=(\frac{2}{2},\frac{14}{2})=(1,7)$$

2. Substitute the coordinates of $R$ into the equation of each line to see which one $R$ lies on:

- For line $y=x+6$:

Substitute $x=1$ and $y=7$:
$$7=1+6$$
$$7=7\text{(True)}$$

- For line $y=x+8$:

Substitute $x=1$ and $y=7$:
$$7=1+8$$
$$7=9\text{(False)}$$

- For line $y=x-6$:

Substitute $x=1$ and $y=7$:
$$7=1-6$$
$$7=-5\text{(False)}$$

- For line $y=x-8$:

Substitute $x=1$ and $y=7$:
$$7=1-8$$
$$7=-7\text{(False)}$$

The midpoint $R(1,7)$ lies on the line $y=x+6$.

The correct answer is A. $y=x+6$