Use traces to sketch and identify the surface. 9 x^2-y^2+z^2=0

Name: Problem 12, Chapter 10: Essential Calculus Early Transcendentals
Uploaded: 2020-07-25T03:32:24-08:00
Duration: 4 min 43 s
Channel: Issa Dababneh
Description: Use traces to sketch and identify the surface. 9 x^2-y^2+z^2=0

Use traces to sketch and identify the surface. 9...

Key Concepts

Quadric Surfaces

Quadric surfaces are second-degree surfaces in three-dimensional space defined by a general second-degree polynomial equation in x, y, and z. They include various common surfaces such as ellipsoids, hyperboloids, paraboloids, and cones. Understanding these surfaces involves studying the roles and signs of squared terms and their coefficients, which determine the surface's shape and orientation.

Traces of Surfaces

Traces, or cross-sections, are the curves obtained by intersecting a surface with a plane. By setting one coordinate constant, one can analyze the resulting two-dimensional conic sections such as ellipses, hyperbolas, and parabolas. This method is a powerful tool for visualizing and sketching the overall structure of a surface in three-dimensional space.

Cones in Quadric Surfaces

Cones are a special type of quadric surface defined by homogeneous second-degree equations. They typically have the property that every cross-section through the vertex produces conic sections, often degenerate, and exhibit symmetry about an axis. Recognizing a cone involves identifying that the equation does not have constant or linear terms and that all terms are of degree two, which implies that the surface is a homogeneous quadratic that tends to open out from a point or vertex.

Transcript

00:01 So in this question, in this question, we're given them 9x squared minus y squared plus z squared is equal to 0.

00:11 And we're asked to use traces in order to identify and sketch the surface.

00:17 So first of all, let's look at this from the xz plane, where our y is going to be a constant.

00:22 So if we set y equal to k, we get 9x squared minus k squared plus z squared is equal to 0.

00:28 So re -arranging, we get 9x squared plus c squared is equal to k squared.

00:34 Dividing by k squared on both sides, we get 9x squared divided by k squared plus z squared divided by k squared plus is equal to 1.

00:45 All right, now another way to write 9 divided by k squared is just as x squared divided by 3 divided by k, and then all of them.

00:55 Sorry, that should be k divided by 3.

01:02 So k divided by 3 and then all of that square.

01:08 And then plus z squared divided by k squared is equal to 1 and you should immediately identify this as an ellipse with its major axis being k divided by 3 and minor axis being simply k.

01:21 So we determined that from the, if we hold y equal to k, from the xz plane, if we look at the xxz plane, we're going to get in the lips.

01:35 So if we look at our surface from the xz plane, we get in the lips.

01:39 What about in the xy plane? well, now we're going to set z equal to k.

01:44 So now we're going to plug in k for z.

01:47 Rearranging, we get y squared minus 9x squared is equal to k squared...

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