Problem 30 - Thomas Calculus

Question

In Exercises \(25-34\) , describe the given set with a single equation or with a pair of equations. The circle of radius 1 centered at \((-3,4,1)\) and lying in a plane parallel to the $$ \text { a. } x y \text { -plane } \quad \text { b. } y z \text { -plane } \quad \text { c. } x z $$

Step-by-Step Solution

Answer

a. \((x+3)^2 + (y-4)^2 = 1\), \(z = 1\); b. \((y-4)^2 + (z-1)^2 = 1\), \(x = -3\); c. \((x+3)^2 + (z-1)^2 = 1\), \(y = 4\).

Step 1: Understand the Problem

We need to find the equation of a circle of radius 1 centered at \((-3, 4, 1)\) in different planes. The circle lies in planes that are parallel to the standard coordinate planes (xy-, yz-, and xz-planes).

Step 2: Circle Equation Parallel to xy-plane

If a circle is parallel to the xy-plane, it means the z-coordinate remains constant at the center's z-coordinate, which is 1. The equation of the circle then becomes: \((x + 3)^2 + (y - 4)^2 = 1\) and \(z = 1\).

Step 3: Circle Equation Parallel to yz-plane

For a circle parallel to the yz-plane, the x-coordinate is constant at \(-3\). The equation of the circle becomes: \((y - 4)^2 + (z - 1)^2 = 1\) and \(x = -3\).

Step 4: Circle Equation Parallel to xz-plane

When the circle is parallel to the xz-plane, the y-coordinate remains constant at 4. Hence, the circle's equation is: \((x + 3)^2 + (z - 1)^2 = 1\) and \(y = 4\).

Key Concepts

Equations of Circles

To understand the equations of circles in 3D coordinate geometry, it's important to recognize that circles are two-dimensional shapes that can reside within various planes in a three-dimensional space. A fundamental formula of a circle centered at a point \(h, k\) in a plane, with radius \(r\), is given by the equation: \((x - h)^2 + (y - k)^2 = r^2\).

In the context of a 3-dimensional coordinate system, these circles can exist parallel to any of the three standard coordinate planes: xy-plane, yz-plane, and xz-plane. When a circle is described as lying parallel to these planes, it means one of the coordinates will remain constant. For this exercise:

• If the circle is parallel to the xy-plane, the z-coordinate is fixed at the value of the center's z-coordinate.
• If it is parallel to the yz-plane, the x-coordinate remains constant.
• Meanwhile, for a circle parallel to the xz-plane, the y-coordinate does not change.

For example, the circle centered at \((-3, 4, 1)\) with a radius of 1, will have these equations depending on its plane of parallelism:

• Parallel to the xy-plane: \((x + 3)^2 + (y - 4)^2 = 1\) and \z = 1\.
• Parallel to the yz-plane: \((y - 4)^2 + (z - 1)^2 = 1\) and \x = -3\.
• Parallel to the xz-plane: \((x + 3)^2 + (z - 1)^2 = 1\) and \y = 4\.

Coordinate Planes

In three-dimensional space, there are three main coordinate planes: the xy-plane, yz-plane, and xz-plane. These planes divide the space and are defined by setting one of the coordinates to zero:

• The xy-plane is characterized by \(z = 0\).
• The yz-plane is at \(x = 0\).
• And the xz-plane lies where \(y = 0\).

These planes are critical in describing positions of objects in 3D geometry, including points, lines, and shapes like circles.

When we say a shape is parallel to a particular plane, it means the shape retains similar symmetry as it would when plotted on that plane, but it resides at a fixed distance from it. In the case of circles, such parallelism means the third coordinate (in which the plane lacks dimension) is constant over the circle, determining its placement relative to that coordinate plane.

This setting allows us to focus purely on the remaining two coordinates, simplifying the problem to a 2D equation relative to the plane. For instance, a circle of radius 1 centered at \((-3, 4, 1)\) lying parallel to the xy-plane holds this position because its z-coordinate remains consistently as \(z = 1\).

Plane Geometry

Plane geometry deals with shapes and sizes that exist in a flat, two-dimensional surface. However, its principles also extend into three-dimensional spaces when considering objects like planes themselves.

A plane in geometry is essentially a flat surface extending infinitely in all directions, defined mathematically by an equation. In three dimensions, this often results from fixing one coordinate across a space, creating a flat region.

For circles within these planes, the same principles apply as those in standard plane geometry: the distance from the center to every point on the circle (the radius) is constant. When considering which coordinate becomes constant, the transition from 3D to a plane-specific geometry helps to simplify complex spatial problems.

Understanding plane geometry helps to manipulate and visualize these geometric shapes effectively, whether in solving for intersection with other planes or determining distances within the 3D space.