Final answer:
The given polar equation represents an ellipse, not a parabola, as it can be rewritten in Cartesian form to fit the definition of an ellipse where the sum of distances from any point on the curve to the foci is constant.
Step-by-step explanation:
The given polar equation r²(2+3cos²(θ))=64 can be analyzed to determine the shape of the curve it represents. It is a conic section, and in polar coordinates, conic sections have the general form r(1+εcos(φ)), where ε is the eccentricity. If ε=1, the conic is a parabola. However, in this equation, we do not have this form, so it's unlikely to represent a parabola. To further analyze the given equation, we expand it to r² + 3r²cos²(θ) = 64. Now we need to see if we can rewrite this in Cartesian coordinates.
Using the polar to Cartesian conversion identities x = r cos(θ) and y = r sin(θ), we attempt to transform the equation. Given that r² = x² + y², we substitute this into the equation and get x² + y² + 3x² = 64, which simplifies to 4x² + y² = 64. This equation does not have the typical form of x = ay² + by + c, which defines a parabola. Instead, it's an ellipse, specifically when rearranged to x²/16 + y²/64 = 1, where the sum of the distances from any point on the curve to the foci is constant. This matches the definition of an ellipse described in your study material.