Problem 6 - Thomas Calculus

Question

In Exercises \(1-16,\) 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

The set of points forms a circle of radius 2 at the plane \(z=-2\).

Step 1: Understanding the Equations

The first equation is \(x^2 + y^2 = 4\). This describes a circle in the XY-plane with a radius of 2, centered at the origin (0,0). The second equation is \(z = -2\), which establishes a fixed height for the entire set of points.

Step 2: Combining the Equations

Since the equation \(x^2 + y^2 = 4\) describes a circle in the XY-plane, and \(z = -2\) dictates that these circles exist at a constant \(z\)-height, we are looking for points on a circular path at \(z = -2\).

Step 3: Visualizing the Geometry

Imagine a circle lying flat in the XY-plane with a radius of 2. This entire circle is then elevated or positioned at \(z = -2\). All points of the set lie on this circle, forming what appears to be a horizontal slice or disc in 3-dimensional space.

Step 4: Geometric Description

The geometric description of the set of points is a circle of radius 2 centered at the point (0, 0, -2) in 3-dimensional space. The circle lies parallel to the XY-plane.

Key Concepts

Circle in 3D Space

Imagine that you have a flat circle that you are holding parallel to the ground. In mathematics, placing a circle in 3D space means that you fix the circle at a certain height or plane. For the equation\( x^{2}+y^{2}=4 \), the circle is centered at the origin, (0,0), with a radius of 2. If we introduce another equation\( z = -2 \), it means that this circle is not at ground level but raised or placed at a height where the z-coordinate is always -2. To visualize:

• The circle lies completely horizontal and parallel at a specific height
• The fixed height is confirmed by the constant z-value, which is -2
• The points of the circle maintain their 2D shape while being located in 3D space

XY-plane

The XY-plane is a fundamental concept in 3D geometry. It is a flat surface where every point is described by an x-coordinate for horizontal position and a y-coordinate for vertical position. When we say the circle described by the equation \( x^2 + y^2 = 4 \) lies in the XY-plane, we mean:

• The circle is fully contained in this flat surface
• All points on this circle have a z-coordinate of 0 unless otherwise specified
• The circle's radius of 2 implies all points are uniformly spread from the center at (0,0) along this plane

The introduction of a z-coordinate of -2 merely shifts the entire plane and circle it harbors to a new 3D location, yet still parallel to the original XY-plane.

Geometric Description

In mathematical terms, giving a geometric description involves explaining the shape and position of a figure in space. For the given equations, the set of points forms a circle. Let's break it down:

• The circle is in the form of a 2D shape, constrained by \( x^2 + y^2 = 4 \)
• Due to \( z = -2 \), every point on this circle is observed at the height of -2 along the z-axis
• This creates a perfectly horizontal circle in 3D space, centered at (0,0,-2)

These points don't just represent a line or a curve, they outline an entire path encircling a central point in space. The geometric description serves to communicate this overall setup succinctly as, 'a circle of radius 2 centered at (0,0,-2) and parallel to the XY-plane.'