Final answer:
The polar equation r²(4sin²(θ)−3)=36 most likely represents a hyperbola due to the behavior of the r² term and its denominator, which suggests two separate branches characteristic of a hyperbola.
Step-by-step explanation:
The polar equation provided is r²(4sin²(θ)−3)=36. To identify the type of conic section it represents, let's manipulate the equation and compare it to standard forms of conic sections. The equation can be rearranged to r² = 36/(4sin²(θ)−3). As a fraction with θ in the denominator, this does not match the standard form for a circle, ellipse, hyperbola, or parabola. Instead, it suggests that for certain values of θ, r might be undefined, which is a characteristic of a hyperbola. This conclusion is based on the fact that for certain angles, the denominator could become zero, which is a condition that leads to two separate branches, as seen in a hyperbola. An ellipse has a general form of x = ay² + by + c. By substituting x = rcos(θ) and y = rsin(θ) into the equation and simplifying, we can see that it matches the general form of an ellipse. Converting this polar equation to Cartesian coordinates is another method to identify the type of conic, but given the initial question, we can deduce that the equation represents a hyperbola without explicit conversion. Therefore, the correct answer would be (C) a hyperbola.