If $e$ is the eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\theta$ is the angle between the asymptotes, then $\cos \frac{\theta}{2}$ is

Through a fixed point $$P(\alpha, \beta), a$$, variable line is drawn to cut the coordinate axes at $$A$$ and $$B$$. The locus of the mid-point of $$A B$$ is