If $(0,0)$, $(4,-3)$ and $(x,y)$ are collinear then

For three points $A(0,0)$, $B(4,-3)$, and $C(x,y)$ to be collinear, the slope of segment $AB$ must equal the slope of segment $BC$ (or $AC$).
$$\text{Slope of }AB=\frac{-3-0}{4-0}=-\frac{3}{4}$$

$$\text{Slope of }AC=\frac{y-0}{x-0}=\frac{y}{x}$$

Equating the slopes:

$$\frac{y}{x}=-\frac{3}{4}⟹4y=-3x⟹3x+4y=0$$

Let us check which option satisfies this linear relation ($3x+4y=0$):

* Option A: $x=8,y=-6⟹3(8)+4(-6)=24-24=0$ (Satisfied)
* Option B: $x=8,y=6⟹3(8)+4(6)=24+24=48\ne 0$
* Option C: $x=4,y=-6⟹3(4)+4(-6)=12-24=-12\ne 0$
* Option D: $x=-8,y=-6⟹3(-8)+4(-6)=-24-24=-48\ne 0$