Curve of intersection of the cone is $z = \sqrt{x^2 + y^2}$ and the ellipsoid is $3x^2 + y^2 + z^2 = 2y$. Find parameterization
Solution:
$z^2 = x^2 + y^2$
$3x^2 + y^2 + z^2 = 2y \implies 4x^2 + 2y^2 - 2y = 0$
$\implies 8x^2 + (2y-1)^2 = 1$ (How?)
So, C lies in the elliptical cylinder given by this equation In XY plane eclipse can be parameterized by
I don't get below:
$x = cos(t)\frac{1}{\sqrt{8}}$
$y = \frac{1}{2}(sin(t)+1)$
$z = [x^2(t) + y^2(t)] = \sqrt{0.5sin(t) - 1/8cos^2(t) + 1/2}$
I don't understand how they get from $4x^2 + 2y^2 - 2y = 0$ to $8x^2 + (2y-1)^2 = 1$
The eliptic cylinder formula is
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
now sure how to use that to parameterize this: $8x^2 + (2y-1)^2 = 1$
any help appreciated