Find a parameterization of the curve of intersection of the paraboloid $z=4x^2 + y^2$ and the parabolic cylinder $y=x^2$
I understand that we need to find a map $M:\mathbb{R} \to \mathbb{R^3}$ such that $\{M(t):t \in \mathbb{R}\} = \{(x,y,z)\in \mathbb{R^3}:y=x^2 \, \wedge \, 4x^2 + y^2=z\}$
Let $x = \sqrt{t}$. Then $y=t$ and $z=4t+t^2$
The we get $$\vec{r}(t)=\begin{pmatrix}\sqrt{t} \\ t^2 \\ 4t+t^2\end{pmatrix}$$ for $\forall \, t \in \mathbb{R}$
I would like to know if this is acceptable because I've seen similiar questions on this website with answers given in trigonometric form. I don't know how to write $\vec{r}(t)$ in terms of $\cos$ and $\sin$.
Best Answer
Square root is not a good idea. You lose half curve
Solve $ \left\{ \begin{array}{l} z=4 x^2+y^2 \\ y=x^2 \\ \end{array} \right. $
$ \left\{ \begin{array}{l} x= t\\ y =t^2\\ z=4 t^2+t^4 \end{array} \right.$
I got this nice image,