Given the equation of a cylinder $x^2+z^2=1,$ describe the curve of intersection between the cylinder and the planes $z=x$ and $y=x$ in the parametric form and the form $F(x,y,z)=0$.
For $z=x$, I have the parametric equations $x=\cos t$, $y=y$ and $z=\cos t$ and then I substituted for $z$ in the equation of the cylinder. I don't know where to go from there.
For $y=x$, I know it's supposed to be an ellipse but I don't know how to write this in the form $F(x,y,z)=0.$ Can anyone give me a hint please?