Find the vector function of the intersection of the surfaces $z={x}^{2}+{y}^{2}$ and 2xy=z

Question

I am trying to find the vector intersection of

$z={x}^{2}+{y}^{2}$ and $2xy=z$

I am usually pretty good at this but I can not figure out which part of the vector function to set to t.

Answer 1

If $(x',y',z')$ is a point in the intersection of these , then we get $(x'-y')^2=0$, or $x'=y'=k$ (say) and $z'=2k^2$. And any point of the form $(k,k,2k^2)$ satisfies both the equations.

Answer 2

solving your two equations we will have $$2xy=x^2+y^2$$ and then we get after dividing by $$xy\neq 0$$ $$2=\frac{x}{y}+\frac{y}{x}$$ this is equivalent to $$\left(\frac{x}{y}-1\right)^2=0$$ it follows $$x=y,z=2x^2=2y^2$$