Question

Sketch the surfaces. $$4 y^{2}+z^{2}-4 x^{2}=4$$

   Sketch the surfaces.
$$4 y^{2}+z^{2}-4 x^{2}=4$$

Thomas Calculus in SI Units

Thomas Calculus in SI Units

George B. Thomas,… 13th Edition

Chapter 12, Problem 43

Instant Answer

Solved by Expert Nick Johnson

06/15/2025

Step 1

To sketch it, we can start by considering the cross-sections of the surface in the $xy$-plane and $xz$-plane. In the $xy$-plane, when $z = 0$, the equation becomes: Show more…

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Sketch the surfaces. $$4 y^{2}+z^{2}-4 x^{2}=4$$

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Key Concepts

Quadric Surfaces

Quadric surfaces are the graphs of second-degree equations in three variables. They include a wide range of surfaces like ellipsoids, hyperboloids, paraboloids, and more. Understanding these surfaces involves recognizing how the coefficients and signs of the squared terms determine the overall shape and classification.

Standard Form Conversion

Converting a given quadratic equation into its standard form is crucial for identifying the type of surface it represents. This often involves dividing by constants or completing the square so that the equation can be compared to well-known canonical forms, thereby revealing the surface's key characteristics.

Cross-sectional Analysis

Cross-sectional analysis involves examining the slices of a three-dimensional surface by intersecting it with planes parallel to the coordinate axes. This method helps in visualizing the surface by revealing familiar curves such as circles, ellipses, or hyperbolas in different planes, which are indicative of the overall structure.

Symmetry

Recognizing symmetry in a quadratic equation greatly aids in sketching the surface. Symmetry with respect to coordinate planes or axes means that certain sections of the surface are mirror images, simplifying the process of visualization and reducing the complexity when drawing the complete surface.

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