91Ó°ÊÓ

First of all, we should convert the polar equation into a Cartesian equation. To do that, we will use the following conversion formulas: \( x = r\cos\theta\) and \(y = r\sin\theta\) Recall that our polar equation is: \(r = \frac{1}{1+2\cos\theta}\) Multiply both sides by \(r(1+2\cos\theta)\): \(r^2 (1+2\cos\theta) = 1\) Now, substitute the conversion formulas: \((x^2+y^2) (1+2\frac{x}{x^2+y^2}) = 1\) Now, we have a Cartesian equation: \((x^2+y^2)(x^2+y^2+2x) = x^2+y^2\)

To plot the curve and visualize how it's generated, we can use the Cartesian equation and plot it using a graphing software/tool or do it manually. It should give us a graph similar to that of a lemniscate.

To show how the curve is generated, we can use arrows and labeled points on the graph. As \(\theta\) increases from \(0\) to \(2\pi\), the curve starts at the origin, and following this sequence, we can place our arrows: 1. Start at \((0,0)\) when \(\theta = 0\) 2. Move rightwards and then upwards for \(\theta\in(0, \pi)\) 3. Reach maximum y-coordinate at \(\theta = \pi/2\) 4. Move downwards and continue leftwards for \(\theta\in(\pi/2, \pi)\) 5. Reach maximum negative x-coordinate (y=0) at \(\theta = \pi\) 6. Move leftwards and then downwards for \(\theta\in(\pi, 3\pi/2)\) 7. Reach minimum y-coordinate at \(\theta = 3\pi/2\) 8. Move upwards and continue rightwards for \(\theta\in (3\pi/2, 2\pi)\) 9. We return to the origin at \(\theta = 2\pi\) Now that we have indicated the direction and points, the curve generation process should be visually clear on the plot.