To find the intersection points, substitute x = k - 2y into the curve equation xy = 6:

\((k - 2y)y = 6\)

\(2y^2 - ky + 6 = 0\)

This is a quadratic in y. For two distinct points, the discriminant must be positive:

\(b^2 - 4ac > 0\)

Here, \(a = 2\), \(b = -k\), \(c = 6\).

\((-k)^2 - 4 \cdot 2 \cdot 6 > 0\)

\(k^2 - 48 > 0\)

\(k^2 > 48\)

\(k < -\sqrt{48} \text{ and } k > \sqrt{48}\)