Find a parametrization, using \cos(t) and \sin(t), of the following curve: the intersection of...

Question:

Find a parametrization, using {eq}\cos(t) {/eq} and {eq}\sin(t) {/eq}, of the following curve: the intersection of the plane {eq}y=3 {/eq} with the sphere {eq}x^2+y^2+z^2=73 {/eq}.

Curves of Intersection


To obtain the curve of intersection of two surfaces, given in Cartesian coordinates, 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 function in order to reduce the number of parameters to one.

Writing the three variables as functions of a single parameter, we obtain a parametric for of a curve in space {eq}\displaystyle x= x(t), y=y(t), z=z(t), t\in\mathbf{R}. {/eq}

Answer and Explanation:


To obtain the vector function of the curve of intersection of the sphere {eq}\displaystyle x^2+y^2+z^2=73 {/eq} with the plane {eq}\displaystyle y=3, {/eq}

we will solve the system {eq}\displaystyle x^2+y^2+z^2=73 \text{ and }y=3. {/eq}

Substituting y into the sphere equation, we will obtain

{eq}\displaystyle x^2+3^2+z^2=73 \text{ and }y=3 \iff x^2+z^2=64 \text{ and } y= 3 \\ \displaystyle \text{ using sine and cosine for } x \text{ and }z:\\ \displaystyle \implies \boxed{\text{ the parametrization is } x=8\cos t, y=3, z=8\sin t, 0\leq t\leq 2\pi}. {/eq}


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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
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