Exercise 25

Let $a$ and $b$ be positive real numbers. You have probably seen the equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ that generates an ellipse, centered at $(h,k)$, with a horizontal axis of length $2a$ and a vertical axis of length $2b$.

1. Write a few sentences to explain why the vector function $\vr$ defined by $\vr(t) = \langle a\cos(t), b\sin(t) \rangle$, $0 \le t \le 2\pi$ is a parameterization of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

2. Find a parameterization of the ellipse $\frac{x^2}{4} + \frac{y^2}{16} = 1$ that is traversed clockwise.

3. Find a parameterization of the ellipse $\frac{(x+3)^2}{4} + \frac{(y-2)^2}{9} = 1$.

4. Determine the $xy$-equation of the ellipse that is parameterized by $$\begin{gathered} \vr(t) = \langle 3 + 4\sin(2t), 1 + 3\cos(2t) \rangle. \end{gathered}$$