Problem 7 - 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. $$x^{2}+z^{2}=4, \quad y=0$$

Step-by-Step Solution

Answer

A circle of radius 2 in the \(xz\)-plane at \(y = 0\).

Step 1: Identify the First Equation

The equation given is \(x^2 + z^2 = 4\). This represents a circle centered at the origin \((0, 0)\) in the \(xz\)-plane with a radius of 2. The reason it's in the \(xz\)-plane is because there is no \(y\) term.

Step 2: Analyze the Second Equation

The second equation is \(y = 0\). This indicates that these points lie on the \(xz\)-plane, meaning the entire geometry is constrained to the plane where \(y\) equals zero.

Step 3: Combine Both Equations

Since the first equation represents a circle of radius 2 in the \(xz\)-plane and the second equation specifies that \(y = 0\), the geometric set is simply the circle. All points satisfying both are those on the circle in the \(xz\)-plane at \(y = 0\).

Step 4: Final Geometric Description

The geometric figure described by these equations is a circle of radius 2 lying entirely in the \(xz\)-plane at \(y = 0\).

Key Concepts

Circle Equation

When dealing with a circle in a coordinate plane, the general equation you might encounter looks like this: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle and \(r\) is the radius.In our context, we have the equation \(x^2 + z^2 = 4\). Notice the absence of a \(y\) term, which suggests that this circle doesn't extend into the \(y\)-dimension. Instead, you find the circle in the \(xz\)-plane.

• The equation represents a circle centered at the origin \((0, 0)\) in this plane.
• The term \(x^2 + z^2 = 4\) tells us the circle has a radius of 2, because \(4\) is \(r^2\).

Furthermore, the simplicity of this equation makes it easy to identify and visualize the circle's geometry in the specified plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves plotting shapes, sizes, and relative positions of figures on a coordinate plane.In this context, our focus is on how equations can describe geometric shapes like circles.

• The equation \(x^2 + z^2 = 4\) represents a circle, situated entirely within the \(xz\)-plane.
• The plane is defined explicitly when you see equations like \(y = 0\), indicating that the shape exists where \(y\) has no value, hence lying flat in the two-dimensional world of the \(xz\)-plane.

With equations, you can precisely map and calculate the points that lie on these geometric figures, understanding their position and size within the coordinate system.

Three-Dimensional Geometry

While two-dimensional planes are often the first point of focus, three-dimensional geometry expands those planes into 3D space. In this universe, every point is described using three coordinates \((x, y, z)\).

• In this exercise, the set of points you are working with are in a 3D space, but constrained to a 2D plane within this space.
• The equation \(y = 0\) effectively "removes" one dimension, confining the geometry to the \(xz\)-plane.

Thus, in three-dimensional geometry, you might often see concepts that seem 2D but exist wholly in a 3D context. Understanding this helps in visualizing and analyzing complex spatial relationships effectively.By exploring equations and constraints in three-dimensional geometry, you learn to navigate beyond flat surfaces and delve into a world where planes, lines, and surfaces intersect and complement one another.