Parametric Equations and Curves — Question 4

Question 4 – Solution

Solution

1. A self-intersection occurs when two different parameter values produce the same point. The graph suggests checking the origin.

2. For the point to lie at the origin, first require $x=0$: $$\cos t=0.$$ On $0\le t\le2\pi$, this gives $$t=\frac{\pi}{2} \quad\text{or}\quad t=\frac{3\pi}{2}.$$

3. Check the $y$-coordinate at both values: $$y\left(\frac{\pi}{2}\right)=\sin\pi=0, \qquad y\left(\frac{3\pi}{2}\right)=\sin3\pi=0.$$ Hence both parameter values produce $(0,0)$.

4. The two visits correspond to different directions. Since $$\frac{dy}{dx}=\frac{2\cos(2t)}{-\sin t},$$ the slopes are $2$ at $t=\pi/2$ and $-2$ at $t=3\pi/2$.

5. Therefore, the curve crosses itself at $$\boxed{(0,0)},$$ and this point is visited exactly twice on the stated interval.