I asked a pretty similar question like this one yesterday but I want to be sure that I'm starting to grasp the concept now so heres another one.
The question goes like this:
a) Parametrize the unit circle centered around origo. Counter-clockwise
b) Parametrize the circle with radius 2 centered around origo. Counter-clockwise
c) Parametrize the unit circle centered around (1,2). Counter-clockwise
d) Parametrize the unit circle centered around origo. Clockwise.
e)Parametrize the ellipse $3x^2 + 4 y^2 = 12 $
f) Change the interval, if you used $ [0, 2 \pi] $ now use $[0,1]$
g) Write an equation for every parametrization
So: My answers are like these:
a) $x^2 + y^2 = 1$
$x=cos(t)$ , $y=sin(t)$
b) $x^2 + y^2 = 2$
$x=\sqrt{2}cos(t)$, $y=\sqrt{2}sin(t)$
c) $(x-1)^2 + (y-2)^2 = 1$
$x=1+cos(t)$ , $y=2+sin(t)$
d) $x = sin(t)$ , $y=cos(t)$ (since it's clockwise)
e) $3(x^2)+4(y^2)=12$ which gives:
$ \frac {x^2} {4} + \frac {y^2} {3} = 1$
$x=2cos(t)$ , $ y = \sqrt 3 sin(t)$
f) I'm pretty clueless here. I only know how to handle parametrization of circles when you know that $(x-a)^2 + (y-b)^2 = d^2$ gives $x = a + dcos(t) $ (and the same for Y but with sin). So, I think the problem is that I don't really understand why that formula works, otherwise I assume it would be pretty simple, right? Any tips here?
g) Pretty clueless here to, does it have anything to do with eliminating the parameter?
Sorry for a big post but I feel like it's important that I really grasp these concepts. I hope a-e is correct so it's basically just f and g I need help with anyway. Although, I guess it's tied with a-e as well since I probably don't know what I'm actually doing there...
Your work is correct.
To parametrize clockwise you can also set $t'=-t \implies \cos t'=\cos t \quad \sin t'=-\sin t$.
For point $f$ just set
$$t=2\pi z \quad z\in[0,1]$$
For point $g$ you are done, eliminating the parameter lead to the cartesian form from which you started.