Final answer:
The region enclosed by the circle x² + y² = 17x in polar coordinates is not a circle centered at the origin, but an offset circle. For an ellipse with a major axis of 16 cm, the semimajor axis is 8 cm, and an eccentricity of 0.8 indicates an elongated shape.
Therefore, the correct answer is: a) A circle with radius 17 centered at the origin.
Step-by-step explanation:
The equation x² + y² = 17x can be converted to polar coordinates where x = r cos(θ) and y = r sin(θ). Substituting these into the equation gives us r² = 17r cos(θ), or r = 17 cos(θ) when r ≠ 0.
The equation r = 17 cos(θ) represents a circle in polar coordinates with a radius of 17/2 and a center at the point (17/2, 0) in Cartesian coordinates. Therefore, the region enclosed by this circle is not necessarily a circle centered at the origin, as one might first think, but rather a circle that is offset from the origin.
Now, answering the question regarding the semimajor axis and eccentricity of an ellipse, if the major axis is 16 cm, then the semimajor axis is half of that, which is 8 cm. Given the high eccentricity of 0.8, we can conclude that this ellipse would be better described as 'very elongated' rather than 'mostly circular'.
Eccentricity measures how much an ellipse deviates from being circular; a value close to 1 indicates a more elongated shape.