If the function $f(x) = \left\{ \begin{array}{l l} -x, & \quad x < 1 \\ a+\cos ^{-1} (x+b), & \quad 1 \leq x \leq 2 \end{array} \right.$ is differentiable at $x=1$, then $\large\frac{a}{b}$ is equal to :

Question

$\begin{array}{1 1} (1) \large\frac{\pi-2}{2} \\ (2) \large\frac{\pi -2}{2} \\ (3) \large\frac{\pi +2}{2} \\ (4) -1-\cos ^{-1} (2) \end{array} $