Given 2 equations $z = x^2 – y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces.
By equating them together, I get $y^2 +xy -1 =0$.
Letting $x=t$, I substitute into the following equation and get $y^2 +ty-1=0$. By solving the equation via completing the squares, I get $y=(\frac{t+\sqrt (t^2+4)}{2})$ or $y=(\frac{t-\sqrt (t^2+4)}{2})$. Is it right to derive that the parametrized equations are:
$x=t$
$y=(\frac{t+\sqrt (t^2+4)}{2})$ or $y=(\frac{t-\sqrt (t^2+4)}{2})$
Best Answer
Firstly, there is a sign error in your computation ; it should be
$$y=\frac{\color{red}{-}t\pm\sqrt{t^2+4}}{2}$$
with a minus sign in front of $t$.
Besides, as your curve is a space curve, you need a third equation $z=...$ expressed too as a function of parameter $t$. this equation will be obtained by plugging the expressions of $x$ and $y$ you have obtained in either of the two surface equations.
In order to have a concrete understanding of the result, here is a graphical representation of the two surfaces which both are 2 kinds of Hyperbolic Paraboloids (http://mathworld.wolfram.com/HyperbolicParaboloid.html) and their intersection curve