On a unit circle with center $O$, $AB$ is an arc with the central angle $\alpha<90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $OB$, $T$ is a point on arc $AB$, and $l$ is the tangent to the circle at $T$. The line $l$ and the angle $AHB$ form a triangle $\Delta$. (a) Prove that the area of $\Delta$ is minimal when $T$ is the midpoint of arc $AB$. (b) Prove that if $S_\alpha$ is the minimal area of $\Delta$ then the function $\frac{S_\alpha}\alpha$ has a limit when $\alpha\to0$ and find this limit.