Answer
(a) $x = acos \space t$ , $y = bsin \space t$, $0 \leq t \leq 2 \pi$
(b) $a = 3 \space and \space b = 1$
$$x = 3cos \space t | y = sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 2$
$$x = 3cos \space t | y = 2sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 4$
$$x = 3cos \space t | y = 4sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 8$
$$x = 3cos \space t | y = 8sin \space t | 0 \leq t \leq 2\pi$$
(c) As $b$ increases, the ellipse gets larger by the y-axis, getting stretched vertically.
Work Step by Step
(a)$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Example 2 gives the parametric equations for a circle:
$$x = cos \space t$$ $$y = sin \space t$$ $$0 \leq t \leq 2\pi$$
In order to eliminate the unkown $t$ in the first equation, we must put $a$ before $cos \space t$ and $b$ before $sin \space t$:
$$x = acos \space t$$ $$y = bsin \space t$$ $$0 \leq t \leq 2\pi$$
So, when we substitute the values into the first equation:
$$\frac{a^2cos^2t}{a^2} + \frac{b^2sin^2t}{b^2} = 1$$ $$cos^2t + sin^2t = 1$$ $$1 = 1$$
(b) Just substitute the values into the parametric equations:
$a = 3 \space and \space b = 1$
$$x = 3cos \space t | y = sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 2$
$$x = 3cos \space t | y = 2sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 4$
$$x = 3cos \space t | y = 4sin \space t | 0 \leq t \leq 2\pi$$
$a = 3 \space and \space b = 8$
$$x = 3cos \space t | y = 8sin \space t | 0 \leq t \leq 2\pi$$
Put the parametric equations in a graphing calculator.
(c)
As we can see on the graph, as $b$ increases, the ellipse gets larger by the y-axis, getting stretched vertically.
