Use \cos(t) and \sin(t), with positive coefficients, to parametrize the intersection of the...

Question:

Use {eq}\cos(t) {/eq} and {eq}\sin(t) {/eq}, with positive coefficients, to parametrize the intersection of the surfaces {eq}x^2+y^2=4 {/eq} and {eq}z=6x^2 {/eq}.

Parametric Curves in Space

A parametric curve in space is given as a vector function of a single prameter {eq}\displaystyle \mathbf{r}(t)=\langle x(t), y(t), z(t) \rangle, t\in\mathbf{R}. {/eq}

To obtain the curve of intersection of two surfaces, we will solve the system of equations with the two equations of the surfaces.

If the surfaces involve a sum of squares of two variables we will involve sine and cosine functions in order to reduce the number of parameters to one.

To find the intersection between the circular cylinder {eq}\displaystyle x^2+y^2=4 {/eq} and the parabolic cylinder {eq}\displaystyle z=6x^2 {/eq}

we will solve the system of the two equation, {eq}\displaystyle x^2+y^2=4 \text{ and } z=6x^2. {/eq}

For this, we will use sine and cosine for x and y, because they appear as a sum of squares.

{eq}\displaystyle \boxed{ x=2\cos t, y=2\sin t, \text{ and } z=24\cos^2 t, 0\leq t\leq 2\pi}. {/eq}