1. **Problem statement:** Find the coordinates of the point that divides the points $P(8,9)$ and $Q(-7,4)$ internally in the ratio $2:3$ and externally in the ratio $4:3$. 2. **Formula for internal division:** If a point $R$ divides the segment $PQ$ internally in the ratio $m:n$, then $$R = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$$ where $P=(x_1,y_1)$ and $Q=(x_2,y_2)$. 3. **Formula for external division:** If a point $S$ divides the segment $PQ$ externally in the ratio $m:n$, then $$S = \left( \frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n} \right)$$ 4. **Calculate internal division point:** Given $P(8,9)$, $Q(-7,4)$, ratio $2:3$ (so $m=2$, $n=3$), $$x = \frac{2 \times (-7) + 3 \times 8}{2+3} = \frac{-14 + 24}{5} = \frac{10}{5} = 2$$ $$y = \frac{2 \times 4 + 3 \times 9}{2+3} = \frac{8 + 27}{5} = \frac{35}{5} = 7$$ So the internal division point is $(2,7)$. 5. **Calculate external division point:** Given ratio $4:3$ (so $m=4$, $n=3$), $$x = \frac{4 \times (-7) - 3 \times 8}{4-3} = \frac{-28 - 24}{1} = -52$$ $$y = \frac{4 \times 4 - 3 \times 9}{4-3} = \frac{16 - 27}{1} = -11$$ So the external division point is $(-52,-11)$. **Final answers:** - Internal division point: $(2,7)$ - External division point: $(-52,-11)$