What is the locus of the center of circles passing through the origin $(0,0)$ with fixed radius?

Let $\alpha$ be a real number. What is the total number of distinct point(s) of intersection between the parabola $y=x^2+4 x \sin \alpha+6$ and the pair of lines $y^2=1$ ?

Let $L$ be a straight line passing through the origin, and it makes angles $\alpha, \beta$ and $\gamma$ with the positive $X, Y$ and $Z$-axes, respectively. What is the value of $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma$ ?