Q$3.(4$ points$)$ Consider the two surfaces $S_1,S_2$ in $R^3$ defined by the following equations in cylindrical polar coordinates:

$S_1$:$,r^2=z(1-z)$

$S_2$:$,r=z$

where $>0$ is fixed.

a$)(2$ points$)$ Show that the intersection of $S_1$ and $S_2$ away from the origin is a circle, and determine its radius.

b$)(2$ points$)$ By expressing the cylindrical coordinate functions $(r,,z)$ in terms of the spherical coordinate functions $(,,),$ express the surfaces $S_1$ and $S_2$ using equations in the spherical coordinates $(,,).$