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:
\[
R = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

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

\[
R = \left( \frac{3 + (-1)}{2}, \frac{5 + 9}{2} \right) = \left( \frac{2}{2}, \frac{14}{2} \right) = (1, 7)
\]

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 \( y = 7 \):
\[
7 = 1 + 6
\]
\[
7 = 7 \quad \text{(True)}
\]

- For line \( y = x + 8 \):

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

- For line \( y = x - 6 \):

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

- For line \( y = x - 8 \):

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

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

The correct answer is A. \( y = x + 6 \)