You're correct!
O(origin) and B(2,3,-4) are on same side of the plane and A(1,-2,3) is on the opposite side of the plane.
Let us assume that the rod AB meets the plane at point X, and the line joining OB is extended to a point Y such that Y lies on the given plane.
If you try to draw it, you would find that the shadow is the line segment joining X and Y.
Now everything is simplified...we shall use vectors to find X and Y.
The general point on the line OB is represented as:
$$\vec{r}=\lambda(2,3,-4)$$
If we put this point on the equation of the plane, we will get the point of intersection of the line OB and the given plane, i.e. the point Y.
Thus,$$2\lambda+3\lambda-8\lambda=1$$
$$\implies\lambda=\frac{-1}{3}$$
Putting the value of $\lambda$ we get $\text{Y}=(-\frac{2}{3},-1,\frac{4}{3})$.
Similarly, with the general equation of line AB: $$\vec{r}=(1,-2,3)+\mu(2,3,-4)$$
Put this point on the equation of plane and we will get the coordinates of the point X.
Thus, $$1+2\mu-2+3\mu+6-8\mu=1$$
$$\implies \mu=\frac{4}{3}$$
Putting $\mu$ in the general point of AB, we get $\text{X}=(\frac{11}{3},2,\frac{-7}{3})$.
Now simply find the distance between X and Y to obtain the length of the shadow.