Find an equation in rectangular coordinates for the surface represented by the cylindrical equation

Question

The given is $$r=6\sin(\theta)$$ Multiplying both sides by $r$ yields $$r^2=6r\sin(\theta)$$ It follows such that $$r^2=6y$$ $$x^2+y^2=6y$$ $$x^2+y^2-6y=0$$ Completing the square $$x^2+y^2-6y+36-36=0$$ $$x^2+(y-6)^2=36$$ I followed the text book example bout the answer isn't correct. I might have messed up with completing the square.

Answer 1

To complete the square you want to write $$ x^2+y^2-6y+9 - 9 = x^2+(y-3)^2 - 9 $$ since $(y-3)^2=y^2-6y+9$.