Final answer:
The polar curve r = 6/(2-sinθ) generally represents a limaçon, which is a type of conic section. The curve does not have typical vertices, center, or major axis as defined for standard ellipses or hyperbolas. To sketch it, one must consider variations in r as θ varies.
Step-by-step explanation:
The polar curve given by r = 6/(2-sinθ) represents a conic section. To identify the type of conic, we can rearrange the equation into the standard form of a conic section in polar coordinates, which is r = ed/(1 - e cos(θ)), where e is the eccentricity and d is the directrix. The given equation could be thought of having e = 1 and d = -6, after identifying the negative sign that would come from the term (-e cos(θ)).
However, we must be cautious because conics defined in this way usually assume e cos(θ), without a negative sign, thus our conic is not standard and its nature should be interpreted with this in mind. In this case, this curve represents a limaçon, which can resemble various shapes depending on the values of e and d, including cardioids and loops.
To investigate further, it's typically easier to analyze such curves by converting to Cartesian coordinates, but as this question primarily concerns properties that can be identified directly from the polar equation, we will focus on those. The vertices and the center are not standard notions for limaçons as they would be for ellipses or hyperbolas.
However, limaçons do not have a major axis as is standardly defined for ellipses. Because the question seems to conflate different types of conics, the intended object of study might be mistranslated, but under the assumption of a limaçon, such characteristics don't apply.
Nevertheless, for illustrative purposes, in the context of an ellipse constructed by the provided method using tacks and string, the center in polar coordinates would be at the origin (0, 0), and the length of the major axis would be the constant total distance between two points on the ellipse and the foci, essentially the length of the string, denoted by 2a.
To sketch the curve, one would typically loop a graph around these points as per the equation, accounting for variations in r as θ varies. However, since an exact sketch isn't possible here, it would be essential to use graphing software or plot several points manually and smooth the curve between them to illustrate the general shape of the limaçon.