Problem 17 - Advanced Engineering Mathematics

Question

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ (x+1)^{2}+(y-2)^{2}+(z+3)^{2}=0 $$

Step-by-Step Solution

Answer

The locus of points is the single point \((-1, 2, -3)\).

Step 1: Analyze the Given Equation

The given equation is \((x+1)^2 + (y-2)^2 + (z+3)^2 = 0\). This equation represents the squared distances of the point \(P(x, y, z)\) from the point \((-1, 2, -3)\) in three-dimensional space.

Step 2: Understand the Structure of the Equation

In three-dimensional space, an equation of the form \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\) defines a sphere centered at \((a, b, c)\) with radius \(r\). In this problem the equation \((x+1)^2 + (y-2)^2 + (z+3)^2 = 0\) indicates the center is \((-1, 2, -3)\) with a radius of 0.

Step 3: Identify the Locus of Points

Since the squared distance equals 0, the only point \(P\) that satisfies this equation must lie exactly at the center of the sphere, which is the point \((-1, 2, -3)\). This implies there is no actual sphere, but simply a single point.

Key Concepts

Three-dimensional space

Three-dimensional space is a concept that allows for the representation of objects with depth, width, and height. Unlike two-dimensional space, which uses coordinates to describe locations and objects based solely on length and width, three-dimensional space requires an additional coordinate to account for depth. Thus, a point in three-dimensional space is defined using three coordinates:

• The x-coordinate, which indicates horizontal placement,
• The y-coordinate, which specifies vertical placement,
• The z-coordinate, which expresses the point's depth.

Understanding this spatial concept is crucial when working with objects that exist in real life, as it enables accurate modeling of their positions and behaviors. The ability to visualize and manipulate points in three-dimensional space is fundamental to disciplines like physics, engineering, and computer graphics.

Equation of a sphere

The equation of a sphere in three-dimensional space is generally given by the formula: \[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\] Here,

• (a, b, c) represents the center of the sphere.
• r represents the radius.

The terms on the left side of the equality represent the squared distances of any point \(P(x, y, z)\) from the center of the sphere.This geometric equation is a powerful tool for determining the locus of points in space that maintains a constant distance from a central point.In the context of the original exercise,note the equation \((x+1)^2 + (y-2)^2 + (z+3)^2 = 0\).In this case, the structure implies a sphere centered at \((-1, 2, -3)\) with a radius of zero,collapsing the sphere into a single point.

Radius of a sphere

The radius of a sphere plays a key role in defining its size, as it is the distance from the center of the sphere to any point on its surface.In the general equation of a sphere \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\),\(r\) represents this crucial measurement. A sphere with a radius greater than zero will encompass a variety of points spread across a region in space. However, when the radius is exactly zero, the equation collapses into a singular point, as no other points satisfy the required distance of zero from the center.Thus, in the exercise where the equation \((x+1)^2 + (y-2)^2 + (z+3)^2 = 0\) is given, the radius is effectively zero.This simplifies our locus of points from a broad sphere to the specific point \((-1, 2, -3)\). It underscores the concept that a sphere's radius determines not only its size but also the geometry it represents within three-dimensional space.