Final answer:
The polar equation r² = sin(2θ) plots as a lemniscate, which resembles a figure-eight. It does not correspond to any of the given options, making the correct answer 'none of the options'. This type of plot shows the symmetry and periodicity inherent in the sine function.
Step-by-step explanation:
The polar equation given is r² = sin(2θ). To plot this, we need to recognize that the equation represents a relationship between the radius r and the angle θ in polar coordinates. The equation can be interpreted as follows: for a given angle θ, the square of the radius r is equal to the sine of twice that angle. Now, let's plot this step-by-step:
- Start at angle θ = 0 and increase θ incrementally.
- For each angle, calculate sin(2θ) and then find the square root of this value to obtain the radius r (noting that both positive and negative square roots will be considered).
- For each value of r and θ, plot the point in the polar coordinate system.
- Continue this process until you have covered a full circle of θ values from 0 to 2π.
A graph of this equation would show that it is not a function that can be classified as intersecting lines, a circle, or a hyperbola. Instead, it's a type of lemniscate, which resembles a figure-eight or infinity symbol. This is because, for certain θ values, sin(2θ) will be positive, giving us positive r values, while for others, it will be negative, resulting in imaginary radius values - which we would exclude from our real plot. The plot will have points symmetric about the origin, due to the periodic nature of the sine function and the squaring of r.
The correct answer to the question is none of the options given, as r² = sin(2θ) does not plot as two intersecting lines (a), a circle (b), a cardioid (c), or a hyperbola (d).