Identify the surface of the given equation

Question

Identify the surface whose equation in spherical coordinates is given by rho = 4 cos phi

A. Paraboloid

B. Plane parallel to the xz-plane

C. Cone

D. Sphere with center (0, 0, 2) and radius 2

E. Cylinder

F. Plane parallel to the xy-plane

Answer 1

We can simplify this problem by converting to cartesian coordinates: ρ=4cosΦ (Eq. 1)

Relevant coordinate transformations:

z=ρcosΦ, therefore cosΦ=z/ρ

√x²+y²+z²=ρ

Substitute for Φ in Eq. 1

ρ=4z/ρ

Rearrange:

ρ²=4z (Eq. 2)

Substitute for ρ in Eq. 2

(√x²+y²+z²)²=4z

Squaring, x²+y²+z²=4z

Subtract 4z from both sides and complete the square:

x²+y²+z²-4z+(4)=+(4)

Factor to get the form of Sphere with center (0, 0, 2) and radius 2 in cartesian coordinates: D

x²+y²+(z-2)²=4