Question

Find a pair of equations describing a circle of radius 3 centered at (2,4,1) that lies in a plane parallel to the \(x z=\) plane

Short answer

Question: Find the pair of equations describing a circle with radius 3 centered at (2, 4, 1) parallel to the xz plane in 3D space. Answer: The pair of equations for the circle are: 1) (x-2)^2 + (z-1)^2 = 3^2 and 2) y = 4

Step-by-step solution

Write the equation of the circle parallel to the xz plane

The circle lies on a plane parallel to the \(xz\) plane, so the \(y\)-coordinate of the center remains constant, which is 4. The equation of a circle in 3D can be expressed generally as \((x-a)^2 + (z-c)^2 = r^2\). In our case, the circle's center is (2, 4, 1), and its radius is 3. We can replace the variables in the equation and obtain the desired equation as follows: \((x-2)^2 + (z-1)^2 = 3^2\).

Write the constant y-coordinate equation

Since the circle is parallel to the xz plane, we know that the \(y\)-coordinate remains constant. To include this in our system of equations, the second equation is simply \(y=4\).

Combine both equations

Finally, we write down the pair of equations that describe the circle of radius 3 centered at (2, 4, 1) and parallel to the \(xz\) plane: 1. \((x-2)^2 + (z-1)^2 = 3^2\) 2. \(y = 4\)

Key concepts

Circle Equation

A circle equation in 3D can differ slightly from the standard 2D format. In a 2D plane, a circle is represented by

• \((x-h)^2 + (y-k)^2 = r^2\)

where

• \((h, k)\) is the center of the circle
• \(r\) is the radius

However, when we describe a circle in a 3D space, it must lie in a specific plane or be defined in terms of three coordinates: \((x, y, z)\).
A circle in 3D space is defined by a set of equations that restrict its position to a plane and providing its circular structure.
In the context of this exercise, the circle equation shows us the planar slice of a 3D space where the circle lives by maintaining one variable constant. Here, the circle is defined as:

• \((x-2)^2 + (z-1)^2 = 3^2\)

This indicates a circle centered at the point \((2, 1)\) on the \(xz\)-plane with radius 3.

3D Coordinate System

Understanding the 3D coordinate system is crucial for visualizing geometry in three dimensions. In 3D space, each point is defined by three coordinates: \((x, y, z)\).

• The \(x\) coordinate represents horizontal placement.
• The \(y\) coordinate indicates vertical elevation.
• The \(z\) coordinate reflects depth.

In the given exercise, the point \((2, 4, 1)\) represents the center of the circle in 3D space. This system forms a cube-like view, with each axis intersecting at a right angle.
Each coordinate plane (e.g., \(xy\), \(xz\), or \(yz\)) provides a flat surface on which objects can reside or through which they can intersect.
The central role of these planes is showcased in the exercise, where a circle lies in a plane parallel to the \(xz\)-plane, helping us perceive which dimensions change and which remain fixed.

Planes in Geometry

In geometry, planes are flat, two-dimensional surfaces that extend infinitely. They play a vital role in describing positions and shapes in 3D space. A plane can be defined using a point and a normal vector, or by aligning it parallel to one of the coordinate planes (\(xy\), \(xz\), \(yz\)).
For the exercise at hand, the circle lies in a plane parallel to the \(xz\)-plane. This setup involves keeping its \(y\)-coordinate consistent, essentially giving us a simple equation:

• \(y = 4\)

This equation ensures that the circle remains fixed at this height, allowing it to "float" over the \(xz\)-plane at the 4-unit mark.
Planes allow us to sectionalize 3D space, segmenting it into distinct regions where geometric figures can be easily described and analyzed. Understanding planes helps in apprehending how shapes interact and coexist within a three-dimensional world.