We know that a straight line L passes through points $\(\mathrm{P}(4,8)\)$; we need to check from the options that which of them is enough to know if the line also passes through $\(\mathrm{X}(12,16) \& \mathrm{Y}(0,0)\)$.

Let the equation of line $\(L\)$ be $\(y=m x+c\)$, where $\(m\)$ is the slope and $c$ is the $\(y\)$-intercept.

As the line $\(L\)$ passes through $\((4,8)\)$, we get $\(8=m \times 4+c=4 m+c \Rightarrow c=8-4 m\)$, so the equation of line $\(L\)$ comes out to be $\(y=m x+8-4 m\)$.

If we get one more point passing through line $\(L\)$ or the value of its slope (i.e. m ), the equation can be found.

Now, checking from the options, we get

(A) The Line $\(L\)$ intersects the straight line $\(M\)$ whose equation is $\(10 x+20 y=0\)-$ which is not sufficient to find the equation of line $\(L\)$.

(B) The Line $\(L\)$ passes through the point $\((2,6)\)$ - using this point we can find the equation of line L and then check if the points $\(\mathrm{X}(12,16) \& Y(0,0)\)$, pass through $\(L\)$ or not.

(C) In the equation of the straight line $L$, the constant term is 0 - which means $\(8-4 \mathrm{~m}=0\)$, so we get $\(\mathrm{m}=2\)$. Thus the equation comes out to be $\(\mathrm{y}=2 \mathrm{x}\)$, so we can check if points $\(X\)$ and $\(Y\)$ lie on the line or not. Hence (C) is sufficient.

(D) The line $\(L\)$ is parallel to the line $\(Q\)$ whose equation is $\(4 x+8 y+12=0\)$ - the equation of line Q \& line $\(L\)$ are parallel to each other, so they have the same slope. Since equation of line $\(Q\)$ gives the value of $\(m\)$, the equation of line $\(L\)$ can be found, so is sufficient.

Hence options (B), (C) \& (D) are true.