Answer
(a) $(x-4)^2+(y+2)^2=25$
(b) circle, center $(4,-2)$, radius$=5$.
See graph.
Work Step by Step
(a) Use the relations $r^2=x^2+y^2, r\ sin\theta=y, r\cos\theta=x, tan\theta=\frac{y}{x}$, we have
$r^2+4r\ sin\theta-8r\ cos\theta=5\Longrightarrow x^2+y^2+4y-8x=5 \Longrightarrow (x-4)^2+(y+2)^2=5+16+4$ or $(x-4)^2+(y+2)^2=25$
(b) We can identify the equation as a circle with center $(4,-2)$, radius$=5$ (rectangular system). See graph.
