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Chapter 1: Problem 30

Find the trace of the given quadric surface in the specified plane of coordinates and sketch it. $$ x^{2}+z^{2}+4 y=0, x=0 $$

Short Answer

Expert verified
The trace of the surface in the plane \(x = 0\) is a downward-opening parabola given by \(y = -\frac{z^2}{4}\) in the yz-plane.

Step by step solution

01

Substitution

We are given the equation of the quadric surface \(x^2 + z^2 + 4y = 0\) and the plane \(x = 0\). To find the trace, we substitute \(x = 0\) into the equation. This simplifies the equation to \(z^2 + 4y = 0\).
02

Solve for y

With the simplified equation \(z^2 + 4y = 0\), solve for \(y\). Rearrange the equation to get \(4y = -z^2\) and then divide by 4 to find \(y = -\frac{z^2}{4}\).
03

Sketch the trace

The equation \(y = -\frac{z^2}{4}\) represents a parabola in the yz-plane, opening downwards. When sketching, treat \(z\) as the horizontal axis and \(y\) as the vertical axis, plotting the vertex at the origin (0,0) and noting the downward opening orientation.

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Key Concepts

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

Trace of a Surface
When dealing with quadric surfaces, one concept that may arise is the 'trace' of a surface. This refers to the intersection of a given surface with a particular plane. Finding the trace helps simplify our understanding of the surface's geometry, allowing us to see cross-sections. To find the trace, we substitute the equation of the plane into the surface equation. For example, with the quadric surface equation \(x^2 + z^2 + 4y = 0\) and the plane \(x = 0\), we substitute \(x = 0\) into the equation, resulting in \(z^2 + 4y = 0\). This gives us a simpler equation that represents the two-dimensional trace within the specified plane.
Coordinate Planes
Coordinate planes are fundamental in visualizing geometry in a three-dimensional space. There are three primary planes: the xy-plane, yz-plane, and xz-plane. Each coordinate plane is defined by setting one of the three variables (x, y, or z) to zero.
  • **xy-plane**: Here, we set \(z = 0\) to observe the relationship between \(x\) and \(y\).
  • **yz-plane**: To explore this plane, we set \(x = 0\). Our exercise utilizes this plane by substituting \(x = 0\) in the quadric surface equation.
  • **xz-plane**: This plane is analyzed by setting \(y = 0\), focusing on the interactions between \(x\) and \(z\).
By understanding these planes, we gain insight into how a complex geometric surface behaves through cross-sections, forming different shapes based on the intersections.
Parabola in yz-plane
In our provided exercise, the trace of the quadric surface in the yz-plane is a parabola. The resulting equation from the substitution \(x = 0\) into the quadric surface is \(y = -\frac{z^2}{4}\). This equation describes a downward-opening parabola. Here are some key characteristics of this shape:
  • The vertex of the parabola is at the origin point \((0,0)\) in the yz-plane.
  • The axis of symmetry is along the z-axis.
  • Since \(-\frac{z^2}{4}\) affects the stretching, it appears wider compared to having a 1 or -1 coefficient.
When sketching this parabola, the z-axis serves as the horizontal reference, while the y-axis is the vertical. Understanding parabolas in coordinate planes, particularly the yz-plane, aids in visual geometric interpretations of surface traces.

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