Parametric Equation for the intersection of a curve

Question

I am trying to find the parametric equation for the cylinder x^2+z^2=4 and the plane through the points (-1,3,0), (1,2,0), and (0,2,2).

I obtained 2x + 4y + z = 10. As the equation of the plane through the points, but I am having trouble finding the parametric equations for the ellipse.

I know the intersection of the cylinder and the plane is an ellipse!

Answer 1

To find parametric equation of the intersection curve of cylinder $x^2+z^2=4$ and plane $2x+4y+z = 10$, we can use polar coordinates.

$x = 2 \cos\theta, z = 2 \sin\theta$

$y = \displaystyle \frac{10-2x-z}{4} = \frac{5-2\cos\theta-\sin\theta}{2}$

So parametric equation of the intersection curve is

$(2\cos\theta, \frac{5-2\cos\theta-\sin\theta}{2}, 2 \sin\theta), \ 0 \leq \theta \leq 2\pi$.