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Chapter 1: Problem 90

For the following exercises, find the traces for the surfaces in planes \(x=k, y=k\), and \(z=k .\) Then, describe and draw the surfaces\(9 x^{2}+4 y^{2}-16 y+36 z^{2}=20\)

Short Answer

Expert verified
The surface is an ellipsoid with elliptical traces in each coordinate plane.

Step by step solution

01

Find traces in plane x = k

Substitute \(x = k\) into the original equation \(9x^2 + 4y^2 - 16y + 36z^2 = 20\). The equation becomes \(9k^2 + 4y^2 - 16y + 36z^2 = 20\). This describes an ellipse in the \(yz\)-plane when solved for given values of \(k\).
02

Find traces in plane y = k

Substitute \(y = k\) into the original equation \(9x^2 + 4y^2 - 16y + 36z^2 = 20\). The equation becomes \(9x^2 + 4k^2 - 16k + 36z^2 = 20\). This describes an ellipse in the \(xz\)-plane for different \(k\).
03

Find traces in plane z = k

Substitute \(z = k\) into the original equation \(9x^2 + 4y^2 - 16y + 36z^2 = 20\). The equation becomes \(9x^2 + 4y^2 - 16y + 36k^2 = 20\). This describes an ellipse in the \(xy\)-plane for varying \(k\).
04

Describe the Surface

The traces found in Steps 1 through 3 are ellipses in each coordinate plane, suggesting that the surface is an ellipsoid. The given equation of the surface should be rewritten into the standard form to thoroughly confirm it. Simplifying and completing the square, the surface is an ellipsoidal shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Traces in Planes
Understanding traces in planes is essential when working with surfaces in analytic geometry. A trace is the intersection of a surface with a plane, and these traces help visualize and understand the surface's geometry.
To find the trace, you substitute a constant value for one of the variables in the equation that describes the surface. It effectively "slices" the surface along that plane and shows the curve of intersection.
  • In the plane where the x-coordinate is constant ( [x=k] ), substituting x into the surface equation simplifies it to a two-variable equation, revealing the type of curve formed on the [yz] -plane.
  • Similarly, for the plane where the y-coordinate is constant ( [y=k] ), the trace can be found on the [xz] -plane.
  • Finally, when the z-coordinate is held constant ( [z=k] ), the resulting trace shows up on the [xy] -plane.
Each of these traces can help identify the overall shape of the surface. For example, if all traces are ellipses, it might suggest that the surface is an ellipsoid.
Ellipses
An ellipse is a significant concept in coordinate geometry. It is a curve that surrounds two focal points, where the sum of the distances to the two foci from any point on the line is constant.
This shape is a conic section, which can come from slicing a cone at an angle. Ellipses have various characteristics:
  • Major Axis: The longest diameter of an ellipse passing through its center.
  • Minor Axis: The shortest diameter that also crosses through the center.
  • Foci: Two fixed points on the interior of an ellipse used in its formal definition.
In this context, the intersections or traces with the coordinate planes result in ellipses since the quintessential terms order the trace equations into the standard form of an ellipse. Remember, the characteristics of these ellipses provide clues to the three-dimensional shape of the original surface.
Coordinate Geometry
Coordinate geometry, or analytic geometry, uses coordinates to examine geometrical concepts and relationships.
This approach facilitates studying shapes by encoding them with an equation of variables, making it easier to understand geometric transformations and properties.
  • Planes: Defined as two-dimensional surfaces extending infinitely in coordinate geometry.
  • Axes: The x, y, and z axes form the framework of three-dimensional coordinate systems.
  • Shapes: These include circles, ellipses, parabolas, and hyperbolas often described by quadratic equations.
In exploring ellipsoids, coordinate geometry helps find characteristics of the shape through its traces, exploring how it cuts through the fundamental planes (such as [x=k, y=k, z=k] ) in a systematic way.

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Most popular questions from this chapter

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