Step-by-Step Solution
Key Concept: The chord of a hyperbola with midpoint $(h,k)$ has equation derived from the difference of the hyperbola equation at two points; the slope condition gives the locus.
Let the midpoint of the chord be $(h, k)$. The equation of the chord is $3hx - 2ky + 4\left(\frac{3}{2}\right) - 6\left(\frac{2k}{3}\right) = 3h^2 - 2k^2 + 4h - 6k$. Simplifying: $(3h + 2)x + (-2k - 3)y = 3h^2 - 2k^2 + 4h - 6k$. The slope of the chord equals $\frac{3h+2}{2k+3} = 2$. The locus of the midpoint is $3x - 4y = 4$.
Correct Answer: 1