Parametric Equation

[uberathlete]

Hi everyone. I'm having problems figuring out how to solve this problem. I don't really know where to start. It goes ...

Find a parametric equation for the curve of intersection of two surfaces z = x^2 - 2y and z = 2x^2 + y^2 .

If anyone can help me on this it'd be much appreciated. Thanks!

 

[soroban]

Hello, uberathlete!

I found a solution . . . but it took some leaps of reasoning.

Find parametric equations for the curve of intersection of two surfaces \(\displaystyle z\:=\:x^2\,-\,2y\) and \(\displaystyle z\:=\:2x^2\,+\,y^2\)

We have: \(\displaystyle \:\begin{array}{cc}[1]\;z\:=\:x^2\,-\,2y\\ [2]\;z\:=\:2x^2\,+\,y^2\end{array}\)

Equate [2] amd [1]: \(\displaystyle \:2x^2\,+\,y^2\:=\:x^2\,-2y\;\;\Rightarrow\;\;x^2\,+\,y^2\,+\,2y\:=\:0\)

Complete the square: \(\displaystyle \:x^2\,+\,(y\,+\,1)^2\:=\:1\)

This is a circle with center (0,-1) and radius 1.

Its parametric equations are: \(\displaystyle \:\begin{Bmatrix}x\:=\:\cos\theta \\ y + 1\:=\:\sin\theta\end{Bmatrix}\;\;\Rightarrow\;\;\begin{Bmatrix}x\:=\:\cos\theta \\ y\:=\:\sin\theta\,-\,1\end{Bmatrix}\)

Substitute into [1]: \(\displaystyle \:z\:=\:\cos^2\theta\,-\,2(\sin\theta\,-\,1)\:=\:1\,-\,\sin^2\theta\,-\,2\sin\theta\,+\,2\:=\:3\,-\,2\sin\theta\,-\,\sin^2\theta\)


The parametric equations are: \(\displaystyle \:\begin{array}{ccc}x\:=\:\cos\theta \\ y\:=\:\sin\theta\,-\,1\\ z\:=\1-\sin\theta)(3+\sin\theta)\end{array}\)