$(8$ points$)$ Consider the intersection curve $c$ of the sphere $x^2+y^2+z^2=1$ and the parabolic cylinder $x-y^2=0.$

$($a$)$ Derive a parameterization of the intersection curve $c($including a suitable parameter interval$).$

$($b$)$ Show that the curve lies on a right circular cylinder $$ whose axis is parallel to the $y-$axis.

$($c$)$ Determine tangent and osculating plane for those points of $c$ which lie in the plane $x=0.$ Show that the osculating plane is tangent to $.$

$($d$)$ Create a visualization $($using any software you are familiar with$).$