Convert to cartesian?

Question

How would I convert $X(t)=\cos(t)a+\sin(t)b$ to cartesian, where $a=(3,3)$ and $b=(-1,1)$. I tried saying $x(t)=3\cos(t)-\sin(t)$ and $y(t)=3\cos(t)+\sin(t)$ but I am stuck on how to remove the $t$.

Answer 1

Try with this: $$x(t)+y(t)=6 \cos t$$ $$-x(t)+y(t)=2 \sin t$$ so that your curve is the ellipse with cartesian equation $$\frac{(x+y)^2}{6^2} + \frac{(x-y)^2}{2^2}=1$$