Problem 6 - Thomas Calculus

Question

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x^{2}+y^{2}=4, \quad z=-2 $$

Step-by-Step Solution

Answer

A circle of radius 2 centered at (0, 0, -2) in the plane z = -2.

Step 1: Understanding the Equation for X and Y

The equation \( x^2 + y^2 = 4 \) represents a circle in the XY-plane. It indicates that for any point that satisfies this equation, the sum of the squares of the x-coordinate and y-coordinate must equal 4. This is typically the equation of a circle centered at the origin \((0,0)\) with a radius \(r\) given by \( r = 2 \).

Step 2: Understanding the Equation for Z

The equation \( z = -2 \) tells us that all points must have a z-coordinate of \(-2\). This represents a horizontal plane parallel to the XY-plane but located at \( z = -2 \).

Step 3: Combining both Equations

To describe the set of points that satisfy both equations, we combine the conditions described in Step 1 and Step 2. This results in a circle of radius 2 on the XY-plane, but specifically in the plane \( z = -2 \).

Step 4: Describing the Geometric Shape

Therefore, all points in space that satisfy both equations are located on a circle with radius 2, centered at the point \((0,0,-2)\). The geometric shape is a circle in a plane parallel to the XY-plane at a height of \( z = -2 \).

Key Concepts

Equation of a Circle

The equation of a circle in a coordinate system is a fundamental concept in 3D geometry. It is usually expressed as \[ x^2 + y^2 = r^2 \] where \( r \) is the radius of the circle.
This equation signifies that any point \((x, y)\) on the circle is exactly \( r \) units away from the center, which typically is at the origin \((0,0)\).
In our exercise:

• We're given \( x^2 + y^2 = 4 \). Here, the circle has a radius \( r = 2 \) because the square root of 4 is 2.
• This equation shows a circle centered at the origin, lying perfectly in the XY-plane.

The circle represents all possible points that lie on its circumference, each maintaining a constant distance, \( r \), from the center.

Coordinate System

Understanding the coordinate system is key to visualizing geometric shapes in space. In 3D geometry, we work with three axes: X, Y, and Z.
Each point in this system is described using coordinates \((x, y, z)\), where:

• \( x \) and \( y \) represent positions on the horizontal and vertical axes in the plane.
• \( z \) indicates how far a point extends above or below the plane of \( x \) and \( y \).

In this exercise, the specific conditions of our coordinate system are:

• The circle lies entirely within the XY-plane, defined by \( z = 0 \).
• However, due to the equation \( z = -2 \), the circle is shifted down, translating the entire circle to occur at \((0,0,-2)\) instead.

The coordinate system helps describe how shapes such as circles are placed in three-dimensional space in relation to these axes.

Planes in Geometry

Planes in geometry are flat, two-dimensional surfaces that extend infinitely in all directions within a particular space. In a 3D coordinate system, you can express a plane with an equation like \(z = c\), where \(c\) is a constant representing the plane's height.
The understanding of planes is crucial because:

• They act as "layers" within 3D space on which shapes like lines, circles, and polygons can lie.
• Each plane is defined by the relationship between the X, Y, and Z coordinates.

In our problem:

• The equation \(z = -2\) creates a horizontal plane that is parallel to the standard XY-plane but positioned 2 units below it.
• The circle from our earlier equation is confined to this plane, illustrating how relations in 3D space can position traditional 2D shapes at various depths.

This demonstrates the relationship between complex spatial concepts and simple geometric equations, crucial in fields such as physics and engineering.