If $A$ & $B$ are the points $(-3, 4)$ and $(2, 1)$, then the coordinates of the point $C$ on produced $AB$ such that $AC = 2BC$ are

The condition $A C=2 B C$ means that $C$ divides the line segment $A B$ externally in the ratio $2: 1$. This is because $C$ lies on the produced side of $A B$, and $A C$ is twice as long as $B C$.

Step 2: Apply the section formula

The section formula for external division is:

$C=\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}\right)$

Here:

- $A\left(x_{1}, y_{1}\right)=(-3,4)$,

- $B\left(x_{2}, y_{2}\right)=(2,1)$,

- The ratio $A C: B C=2: 1$, so $m=2$ and $n=1$.

Substitute the values into the formula:

$C=\left(\frac{2 \cdot 2-1 \cdot(-3)}{2-1}, \frac{2 \cdot 1-1 \cdot 4}{2-1}\right)$

Simplify:

$\begin{array}{c} C=\left(\frac{4+3}{1}, \frac{2-4}{1}\right) \\ C=(7,-2) \end{array}$