Problem 5 - University Calculus: Early Transcendentals

Question

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}=4, \quad z=0$$

Step-by-Step Solution

Answer

A circle of radius 2 in the xy-plane (z=0).

Step 1: Analyze the First Equation

The equation \(x^2 + y^2 = 4\) represents a circle in the xy-plane. This circle has its center at the origin (0, 0) and a radius of 2, as any point on the circle satisfies \(x^2 + y^2 = r^2\) where \(r = 2\).

Step 2: Analyze the Second Equation

The equation \(z = 0\) indicates that the z-coordinate of any point is 0. This means the entire set of points lies in the xy-plane, also known as the "z=0 plane".

Step 3: Combine the Equations

By combining \(x^2 + y^2 = 4\) and \(z = 0\), we conclude that the set of points is a circle of radius 2 centered at (0,0,0) in the xy-plane. This circle lies flat on the z=0 plane, which means it forms a horizontal circle in the 3-dimensional space.

Key Concepts

circle in xy-plane

In coordinate geometry, a circle in the xy-plane can be described by the equation \(x^2 + y^2 = r^2\). This equation represents all the points that maintain a constant distance, known as the radius \(r\), from a center point. In this case, the center is often the origin \((0, 0)\). For the given equation \(x^2 + y^2 = 4\), the circle is centered at \((0, 0)\) with a radius of 2, because we assume \(r^2 = 4\). This means every point on the circle is exactly 2 units away from the center point. Understanding this allows us to visualize how circles behave in the xy-plane, confined to what we often call 'two-dimensional space.'

coordinate geometry

Coordinate geometry involves using a system of coordinates to describe the location and shape of geometric figures. In the Cartesian coordinate system, geometric shapes like circles, lines, and parabolas can be defined using equations.

In our case, the equation \(x^2 + y^2 = 4\) can be understood as a strict two-dimensional representation within this system of coordinates. It is focused only on the x and y values, implying that it lies completely on the xy-plane, without involving the z-axis. Coordinate geometry allows us to not only define these shapes precisely but also to analyze their relationships, study transformations, and apply various algebraic methods.”

3D space analysis

When analyzing shapes in 3D space, we introduce an additional dimension: the z-axis. This allows us to interpret figures not just as flat, but as part of a volume in space. The equation \(z = 0\) indicates that a certain shape or collection of points is tethered directly to the xy-plane, essentially acting as a 'slice' of 3D space.

By combining the equation \(x^2 + y^2 = 4\) with \(z = 0\), we analyze this condition as a circle confined to the z=0 plane but in the 3-dimensional context. This means the circle forms a flat disk that lies exactly in the xy-plane, existing within the larger arena of three dimensions, but not venturing into the full depth that a 3D space involves. The analysis of such formations is crucial for understanding spatial structures and visualizing geometric relationships in 3D environments.