$(6$ points$)$ Consider the curve $C_1$ in $R^2$ defined by the following equation in $(x,y)-$coordinates:

$9x^2+16y^2=25$

a$)(1$ point$)$ Write down an equation for the curve using the polar coordinate functions $r$ and $,$ of the form $r=f().$

b$)(1$ point$)$ Use this equation to deduce the minimum and maximum values of the radial function $r$ on the curve.

Now consider the curve $C_2$ in $R^2$ with an $(x,y)-$coordinate parametrization as follows:

$x=cos(t),y=\frac{1}{2}sin(2t)$

c$)(2$ points$)$ Show that the polar coordinate equation $r^2=\frac{cos(2)}{cos({)}^4}$ defines the same curve.

d$)(2$ points$)$ Find the subset of values of $$ between $0$ and $2$ for which there exists a point on the curve $C_2$ with angle $.$