Calculator active. Consider the polar curve defined by the function $r()=2cos,$ where $0\frac{3}{2}.$ The derivative of $r$ is given by $\frac{dr}{d}=2cos-2sin.$ The figure above shows the graph of $r$ for $0\frac{3}{2}.$

a$.$ Find the area of the region enclosed by the inner loop of the curve.

b$.$ For $0\frac{3}{2},$ find the greatest distance from any point on the graph of $r$ to the origin. Justify your answer.

c$.$ There is a point on the curve at which the slope of the line tangent to the curve is $\frac{2}{2-}.$ At this point, $\frac{dy}{d}=\frac{1}{2}.$ Find $\frac{dx}{d}$ at this point.