We need to check that which of the given options is sufficient to find the slope of line $\(L\)$.

(A) the area of the triangle formed by three points on the straight line $\(L\)$ is 0 - as points on line are not given, slope cannot be found.

(B) One point on the line $\(L\)$ is $\((x, 0)\)$ and another point on the line $\(L\)$ is $\((0, y)\)-$ if two points are given, slope can be found, so is sufficient.

(C) the straight line $L$ is parallel to a line whose equation is $\(2 x+3 y=0\)$ - parallel lines have same slope, so the slope of line $\(L\)$ should be same as the slope of line $\(2 x+3 y=0\)$ i.e. $\(-2 / 3\)$, so is sufficient.

Hence options (B) \& (C) are correct.