<s>[INST] Find the parametrization of the curve resulting from intersection of a function and a curve I have the following function $f(x,y) = 2-x^2-4y^2$ and the surface $2x+4y+z-1 = 0.$
How do i go about finding the parametrization of the curve resulting from intersection of these surfaces? I see that $f(x,y)$ is the equation of an ellipsoid. I have tried to set $ f(x,y) = z$ and go from there but i cant seem to find any parameterization.
 [/INST] We have


*

*$z=2-x^2-4y^2$

*$z=1-2x-4y$


then
$$2-x^2-4y^2=1-2x-4y\iff x^2-2x-1+4y^2-4y=0 \\\iff(x-1)^2+(2y-1)^2=3$$
thus we can take


*

*$x=\sqrt 3\cos \theta +1$

*$y=\frac{\sqrt 3}2\sin \theta +\frac12$

*$z=1-2x-4y=1-2\sqrt 3\cos \theta -2-2\sqrt 3\sin \theta -2=-3-2\sqrt 3(\cos\theta+\sin \theta)$

</s>