Cylindrical Coordinates — Question 4

Question 4

Describe the surface $r=4\cos\theta$ in Cartesian coordinates.

Tasks

1. Convert without dividing by $r$.

2. Identify the surface and its axis.

3. State its intersections with the coordinate axes.

Question 4 – Solution

Strategy. Multiply by $r$ and use $r^2=x^2+y^2$ and $r\cos\theta=x$.

Step 1: Convert From $r=4\cos\theta$, $$r^2=4r\cos\theta\quad\Longrightarrow\quad x^2+y^2=4x.$$ Completing the square, $$\boxed{(x-2)^2+y^2=4}.$$

Step 2: Identify Since $z$ is unrestricted, this is a circular cylinder of radius $2$ whose axis is the vertical line $x=2,y=0$.

Step 3: Axis intersections In the $xy$-plane it meets the $x$-axis at $(0,0,0)$ and $(4,0,0)$ and the $y$-axis only at the origin. More generally, the cylinder contains the vertical lines $(0,0,z)$ and $(4,0,z)$.