How to parameterize an ellipse?

Question

I need to parameterize the ellipse $\frac{x^2}{2}+y^2=2$, so this is how I proceed:

I know that $a=2$ and $b=1$ (where $a$ and $b$ are the axis of the ellipse), so I parameterize as: \begin{cases} x=a\cos(t)\\ y=b\sin(t) \end{cases} and I get \begin{cases} x=2\cos(t)\\ y=\sin(t) \end{cases} but if I plot the parameterized curve and the equation those are not the same, I think that the cause it's the $=2$ in the equation but I don't know how to put it into the parameterized ellipse, can someone help me?

Answer 1

Divide by 2 and write the denominator of the $y$ term as $(\sqrt2)^2$: $$\frac{x^2}{2^2}+\frac{y^2}{(\sqrt2)^2}=1$$ This gives the correct parametrisation: $$x=2\cos t\qquad y=\sqrt2\sin t\qquad t\in[0,2\pi]$$

Answer 2

I know that $a=2$ and $b=1$ (where $a$ and $b$ are the axis of the ellipse)

Nope, they are NOT the semi-major and semi-minor axis. The equation needs to be reduced to

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

Notice the $1$ on the right side. So you need to divide your equation to get

$$ \frac{x^2}{4} + \frac{y^2}{2} = 1 $$