# Thread: Equation of curve of intersection (R3)

1. ## Equation of curve of intersection (R3)

What is the equation of the curve of intersection of:

$z = x^2 + y^2$ and

$2y + z = 0$

(Refer to attachment for graph of both equations).

2. Please ignore the previous attachment.

Refer to the correct attachment below.

3. Originally Posted by SyNtHeSiS
What is the equation of the curve of intersection of:

$z = x^2 + y^2$ and

$2y + z = 0$
.
What have you tried yourself? Do you understand that, because this is a one dimensional curve in three dimensions, you will need to find three equations for x, y, and z in terms of a siingle parameter?

Since $z= x^2+ y^2$, that second equation is $2y+ z= 2y+ x^2+ y^2= 0$ which is the same as $x^2+ y^2+ 2y+ 1= 1$ or $x^2+ (y+ 1)^2= 1$. That is, the projection of that curve down to the xy-plane is the circle of radius 1 with center at (0, -1).

Do you know how to parameterize such a circle? Use either $z= x^2+ y^2$ or $z= -2y$ to complete the parametric equations for this curve.