Let's solve the problem step-by-step.

Given Information:

1. Equation of the line $L$: $x - y = 3$
2. Equation of the curve $C$: $3x^2 - y^2 + xy = 9$

The line $L$ and the curve $C$ intersect at two points $P$ and $Q$. We need to:

1. Find the coordinates of points $P$ and $Q$.
2. Calculate the midpoint of $PQ$.

Step 1: Express $y$ in Terms of $x$ from the Line Equation

The line equation is: $x - y = 3$ Rearrange to express $y$ in terms of $x$: $y = x - 3$

Step 2: Substitute $y = x - 3$ into the Curve Equation

Now substitute $y = x - 3$ into the curve equation: $3x^2 - y^2 + xy = 9$ Substitute $y = x - 3$: $3x^2 - (x - 3)^2 + x(x - 3) = 9$

Step 3: Simplify the Equation

Expanding $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$ So, $3x^2 - (x^2 - 6x + 9) + x(x - 3) = 9$ $3x^2 - x^2 + 6x - 9 + x^2 - 3x = 9$ Combine like terms: $3x^2 - x^2 + x^2 + 6x - 3x - 9 = 9$