<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] You have


*

*$z = f(x,y)= 2- x^2-4y^2$

*$2x+4y+z-1 = 0 \Leftrightarrow z=1-2x-4y$


It follows
$$1-2x-4y = 2- x^2-4y^2 \Leftrightarrow x^2-2x+4y^2-4y = 1$$
Square completion gives
$$(x-1)^2-1 +(2y-1)^2-1 = 1 \Leftrightarrow (x-1)^2 +(2y-1)^2 = 3 \Leftrightarrow$$ $$ \left( \frac{x-1}{\sqrt{3}} \right)^2 + \left( \frac{2y-1}{\sqrt{3}} \right)^2 = 1$$
Now, set 
$$\frac{x-1}{\sqrt{3}} = \cos \phi,\; \frac{2y-1}{\sqrt{3}} = \sin \phi$$
Solve for $x$ and $y$ and plug it into one of the given equations to get a parametrization for $z$.
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