1. **Problem:** Find the coordinates of point C such that $AC = 2BC$ where $A=(-3,4)$ and $B=(2,1)$. 2. **Formula:** If $C$ divides the segment $AB$ in the ratio $m:n$, then coordinates of $C$ are given by $$C=\left(\frac{m x_B + n x_A}{m+n}, \frac{m y_B + n y_A}{m+n}\right)$$ 3. **Understanding the problem:** Here, $AC = 2BC$ means $C$ divides $AB$ internally in the ratio $AC:CB = 2:1$. Since $C$ lies between $A$ and $B$, the ratio $m:n = 2:1$ where $m$ corresponds to segment $BC$ and $n$ corresponds to segment $AC$. But since $AC=2BC$, $C$ divides $AB$ in ratio $2:1$ starting from $A$ to $B$. So, $C$ divides $AB$ in ratio $2:1$ from $A$ to $B$. 4. **Calculate coordinates of $C$:** Using the section formula with $m=2$ and $n=1$, $$C_x = \frac{2 \times 2 + 1 \times (-3)}{2+1} = \frac{4 - 3}{3} = \frac{1}{3}$$ $$C_y = \frac{2 \times 1 + 1 \times 4}{2+1} = \frac{2 + 4}{3} = \frac{6}{3} = 2$$ 5. **Final answer:** The coordinates of point $C$ are $$\boxed{\left(\frac{1}{3}, 2\right)}$$ Note: The given options do not include this exact answer, so the correct coordinates are $(\frac{1}{3}, 2)$ based on the problem statement and formula.