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Chapter 1: Problem 5

For the following exercises, describe and graph the set of points that satisfies the given equation. $$ (y-1)^{2}+(z-1)^{2}=1 $$

Short Answer

Expert verified
The points form a circle of radius 1, centered at (1, 1) in the yz-plane.

Step by step solution

01

Identify the Type of Equation

The given equation is \((y-1)^{2}+(z-1)^{2}=1\). This resembles the standard form of a circle equation \((x-a)^2 + (y-b)^2 = r^2\), where \(a\) and \(b\) represent the center of the circle and \(r\) is the radius. In this case, the circle is in the \(yz\)-plane.
02

Determine the Center and Radius

From the equation, we can identify that the center of the circle is located at \((y, z) = (1, 1)\). The right-hand side of the equation is \(1\), which represents \(r^2\), where \(r\) is the radius of the circle. Thus, the radius \(r\) is \(\sqrt{1} = 1\).
03

Describe the Set of Points

The set of points described by this equation is a circle centered at \((y, z) = (1, 1)\) in the \(yz\)-plane. The radius of this circle is \(1\). Since \(x\) is not included in the equation, the circle is in the plane parallel to the \(yz\)-plane for any value of \(x\).
04

Graph the Set of Points

To graph the set of points, draw a circle centered at the point \((y, z) = (1, 1)\) within the \(yz\)-plane. The radius of the circle will be 1. This circle will be the same for any coordinate \(x\), creating a cylindrical shape with axis parallel to the \(yz\)-plane.

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Key Concepts

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

Circle Equation
A circle equation in a 2D space, like the one given \((y-1)^{2}+(z-1)^{2}=1\), looks similar to the more familiar circle equation in the form of \((x-h)^{2}+(y-k)^{2}=r^2\). Here, \(h\) and \(k\) are the coordinates of the circle's center, and \(r\) is the radius. In this specific equation, \(h = 1\) and \(k = 1\), hence the center of the circle is at \((y, z) = (1, 1)\).
  • The standard circle equation: \((x-a)^2 + (y-b)^2 = r^2\)
  • Center \((a, b)\) determines location
  • Radius \(r\) defines the size
Understanding this circle equation helps predict where a circle lies and its size. Since this equation only includes variables \(y\) and \(z\), it describes a circle in the \(yz\)-plane with respect to an \(x\) value.
Cylindrical Shape
When you have a circle in a plane, and extend it along a new axis, it forms a cylindrical shape. In our equation \((y-1)^{2}+(z-1)^{2}=1\), the circle lies flat in the \(yz\)-plane and is independent of the \(x\) coordinate. This means for every value of \(x\), from \(-\infty\) to \(\infty\), the circle remains the same. It effectively creates a cylinder that is parallel to the \(x\)-axis.
  • Base circle: Defined in any plane, like \(yz\)
  • Extension: Constant along an axis, such as \(x\)
  • Cylinder: Infinite in one direction, full along others
Visualizing this cylindrical shape helps in understanding how circles translate into 3D objects when extended through another dimension.
Radius in 3D
In 3D geometry, the concept of a radius extends beyond ordinary circles. In our specific problem, the radius \(r\) of the circle in the equation \((y-1)^{2}+(z-1)^{2}=1\) is 1. It serves multiple functions when relating to a cylindrical shape.
  • Determines circle size in any planar slice
  • Fixed in \(yz\)-plane, spans entire cylinder height
  • Helps in calculating surface areas and volumes
By recognizing the radius's role in three dimensions, it becomes easier to grasp the transition from simple 2D geometric figures to their 3D representations. When thinking about a cylinder, consider how a fixed radius determines the potential surface and shape throughout its length.

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Most popular questions from this chapter

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