which of the following statements best describes the graph of r² = a² cos 2θ? choose the correct answer. o…

which of the following statements best describes the graph of r² = a² cos 2θ? choose the correct answer. o a. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = 0 and θ = π. o b. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/3 and θ = 4π/3. o c. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/4 and θ = 5π/4. o d. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/2 and θ = 3π/2.

which of the following statements best describes the graph of r² = a² cos 2θ? choose the correct answer. o a. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = 0 and θ = π. o b. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/3 and θ = 4π/3. o c. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/4 and θ = 5π/4. o d. the graph is a lemniscate. the endpoints of the two loops of the lemniscate occur when θ = π/2 and θ = 3π/2.

Answer

Explanation:

Step1: Recall polar - curve properties

The equation (r^{2}=a^{2}\cos2\theta) represents a lemniscate. For the endpoints of the loops of the lemniscate in polar - coordinates, we set (r = 0).

Step2: Solve for (\theta) when (r = 0)

If (r^{2}=a^{2}\cos2\theta=0), then (\cos2\theta = 0). We know that (\cos\alpha=0) when (\alpha=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}). So, (2\theta=(2n + 1)\frac{\pi}{2}), which gives (\theta=(2n + 1)\frac{\pi}{4}). In the range ([0,2\pi]), when (n = 0), (\theta=\frac{\pi}{4}) and when (n = 1), (\theta=\frac{5\pi}{4}).

Answer:

C. The graph is a lemniscate. The endpoints of the two loops of the lemniscate occur when (\theta=\frac{\pi}{4}) and (\theta=\frac{5\pi}{4}).