Cylindrical Coordinates — Question 3

Question 3 – Solution

Strategy. Replace $x^2+y^2$ by $r^2$ and $x$ by $r\cos\theta$.

Step 1: Convert $$r^2-6r\cos\theta+4z=3,\qquad \boxed{z=\frac{3-r^2+6r\cos\theta}{4}}.$$ For geometric identification, complete the Cartesian square: $$(x-3)^2+y^2=12-4z.$$

Step 2: Identify Equivalently, $$z=3-\frac{(x-3)^2+y^2}{4}.$$ This is a downward-opening circular paraboloid with vertical axis $x=3,y=0$ and vertex $\boxed{(3,0,3)}$.

Step 3: Traces At height $z=c$, the trace is a circle centered at $(3,0,c)$ with radius $2\sqrt{3-c}$, present for $c\le 3$.