15. The line $y=k x+1$ intersects the left branch of the hyperbola $x^{2}-y^{2}=1$ at points $A$ and $B$. The line $l$ passes through the fixed point $P(-2,0)$ and the midpoint of chord $A B$. Find the range of the y-intercept $b$ of $l$.
15. $\left\{\begin{array}{l}y=k x+1, \\ x^{2}-y^{2}=1 \quad(x>0, \\ \triangle=(2 k)^{2}+8\left(1-k^{2}\right)>0 . \end{array}\right.$

Solving, we get $k \in(1, \sqrt{2})$.
Let $N$ be the midpoint of $A B$, then $x_{N}=\frac{k}{1-k^{2}}, y_{N}=\frac{1}{1-k^{2}}$, the equation of line $l$ is $y=\frac{\frac{1}{1-k^{2}}}{-\frac{k}{1-k^{2}}+2} +2$,
Let $x=0$, then $b=y=\frac{2}{-2 k^{2}+k+2}, k \in(1, \sqrt{2})$ and $k \neq \frac{1+\sqrt{17}}{4}$, so $b \in(-\infty,-2-\sqrt{2}) \cup(2,+\infty)$.