# Thread: parameterize curve of intersection

1. ## parameterize curve of intersection

Ellipsoid has equation $\displaystyle x^2+y^2-yz+z^2=1$
It fits inside a cylinder, which if I did the math correctly, is $\displaystyle x^2+y^2-\frac{y^2}{2}+\frac{y^2}{4}=1$ Parameterize the curve of intersection of the ellipsoid with the elliptic cylinder.

So I'm looking for a function that describes where the ellipsoid touches the cylinder?

I'm not sure where to start

2. The equations we are given are:

\displaystyle \begin{aligned} x^2+y^2-yz+z^2&=1\\ x^2+\frac{3}{4}y^2&=1. \end{aligned}

We must parametrize all solutions of this equation as a set of points $\displaystyle (x(t),y(t),z(t))$. Hint: What happens when the second equation is subtracted from the first?

3. ok I get $\displaystyle \frac{1}{4}y^2-yz+z^2=0$
now I believe the next step is to solve for y or z. I am stuck on how to do this. I used a CAS and got z=y/2

4. Originally Posted by superdude
ok I get $\displaystyle \frac{1}{4}y^2-yz+z^2=0$
now I believe the next step is to solve for y or z. I am stuck on how to do this. I used a CAS and got z=y/2
to solve this by hand first multiply the equation by 4 to get

$\displaystyle y^2-4zy=-4z^2$ Now lets complete the square on the left hand side

$\displaystyle y^2-4xy+4z^2=-4z^2+4z^2 \iff (y-2z)^2=0 \iff y=2z$