Area Ratio and Enclosed Region in a Triangle

Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of $\triangle ABC$, $P$ be a point on side $\overline{BC}$, and $\Delta_1$ and $\Delta_2$ be the areas of $\triangle APB$ and $\triangle ABC$, respectively. If $\Delta_1:\Delta_2 = 4:7$, what is the area enclosed by the lines $\overline{AP}$, $\overline{AC}$, and the $x$-axis? The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are coprime integers. Input the value of $\alpha+\beta$.