7. Given the hyperbola $P: \frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ with left and right foci $B, C$, and a point $A$ on $P$, $I$ is the incenter of $\triangle A B C$ and the line $A I$ passes through the point $(1,0)$. If $\overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C}$, then $x+y=$ $\qquad$
Answer: $\frac{3}{4}$
Difficulty: Medium
Test: Hyperbola, Collinear Vectors, Incenter
Analysis: Let $A I \cap B C=D(1,0)$, then $A$ is on the right branch of $P$, and $C(5,0)$. Since $B, D, C$ are collinear, we have $\overrightarrow{A D}=k \overrightarrow{A I}=k x \overrightarrow{A B}+k y \overrightarrow{A C} \Rightarrow k x+k y=1$
Then $\frac{1}{x+y}-1=k-1=\frac{A D}{A I}-1=\frac{I D}{A I}=\frac{B D}{B A}=\frac{D C}{A C}=\frac{B D-D C}{B A-A C}=\frac{10-2 D C}{6}$ $\Rightarrow x+y=\frac{3}{4}$