Problem 12 - Algebra and Trigonometry

Question

\(11-14\) . Find an equation of a sphere with the given radius \(r\) and center \(C .\) $$ r=3 ; \quad C(-1,4,-7) $$

Step-by-Step Solution

Answer

The equation of the sphere is \((x+1)^2 + (y-4)^2 + (z+7)^2 = 9\).

Step 1: Understand the Equation of a Sphere

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\).

Step 2: Identify the Given Values

From the problem, we know the center of the sphere is \((-1, 4, -7)\) and the radius is \(r = 3\). These values correspond to (h, k, l) = (-1, 4, -7) and \(r = 3\).

Step 3: Substitute Values into General Equation

Substitute the center \((-1, 4, -7)\) and radius \(3\) into the general equation of the sphere: \((x+1)^2 + (y-4)^2 + (z+7)^2 = 3^2\).

Step 4: Simplify the Equation

Calculate \(3^2 = 9\) to simplify the equation to \[(x+1)^2 + (y-4)^2 + (z+7)^2 = 9\].

Key Concepts

Center of a Sphere

The center of a sphere is a fundamental point that defines the location of the sphere in a three-dimensional space. It is represented by the coordinates \(h, k, l\). In the context of a sphere, the center is essentially the "middle point" from which every point on the surface of the sphere is equidistant.

To find the center of a sphere from its equation, you need to identify the values of \(h, k, l\). For example, if the equation of a sphere is expressed as \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \), then the center is directly given by \( (h, k, l) \). This is because the equation stems from the definition of distance in three-dimensional space, where each squared term represents the distance from a point on the sphere to the respective center coordinate.

For instance, in the problem given, the center is \((-1, 4, -7)\). This means when you plug in these values in place of \(h, k, \text{and} l\), it tells you exactly how the sphere is positioned relative to the origin of the coordinate system.

Radius of a Sphere

The radius of a sphere is the distance from the center of the sphere to any point on its surface. It is a constant value for any given sphere and is symbolized by \(r\).

For identifying the radius from the general equation of a sphere \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), look at the right side of the equation, which is equal to \(r^2\). To get \(r\), simply take the square root of that value.

In practical terms, the radius measurement is crucial in understanding the size of the sphere. In our given problem, the radius is \(r = 3\). When you square it to fit into the formula, you get \(3^2 = 9\), which appears on the right side of the equation. This shows how the radius dictates the spread of the sphere in three-dimensional space.

General Equation of Sphere

The general equation of a sphere provides a concise mathematical way to describe the sphere using its center and radius. The standard form of this equation is \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \). It brings together the main elements that define a sphere: its center \( (h, k, l) \) and its radius \(r\).

In constructing this equation, each squared term \( (x-h)^2, (y-k)^2, (z-l)^2 \) represents the square of the distance from any point on the sphere to the center along each coordinate axis. The sum of these squared distances equals the square of the radius, completing the sphere's description.

For example, substituting in our specific values of \(h = -1, k = 4, l = -7\), and \(r = 3\) into the equation results in \( (x+1)^2 + (y-4)^2 + (z+7)^2 = 9 \). This equation completely defines the sphere in the three-dimensional plane, specifying both its position and size.