91Ó°ÊÓ

The center of a sphere is a point that is located at the "heart" of the sphere, equidistant from any point on the surface. For spheres represented in three-dimensional Cartesian coordinates, the center is denoted by the coordinates \( (h, k, l) \). This point is crucial because it anchors the entire sphere.

Given the equation of a sphere, identifying the center involves examining the terms with the variables \(x\), \(y\), and \(z\). If the equation is not in the standard form (which we'll discuss soon), you might need to rearrange or complete the square to find these coordinates.

In the example equation \(x^2 + y^2 + z^2 + 4x - 8z + 19 = 0\), the center is derived by transforming the equation into its standard format. By doing so, we identify the center as \((-2, 0, 4)\). This shows how these coefficients in front of \(x\), \(y\), and \(z\) guide us in pinpointing the sphere's center.

The radius of a sphere is the distance from the center to any point on its surface. It plays a vital role in defining the sphere's size and is a constant value across its entire volume.

In a sphere's equation, once centered at \( (h, k, l) \), the radius \(r\) can be calculated using the formula:

• If the sphere's equation is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), the radius \(r\) is the square root of the right-hand side.

For the given exercise, where the equation extends and includes linear terms, the radius is calculated from the expression:

• \( \sqrt{g^2 + f^2 + h^2 - c} \)

Here, \(g\), \(f\), and \(h\) are coefficients derived from the expanded form, assisting in finding the value \(\sqrt{2}\), which is the radius of the sphere in our example.

A sphere's equation can be expressed in what's known as its standard form, providing clarity and simplicity. This form is crucial because it directly provides insights into both the center and the radius.

Written typically as \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), this format indicates:

• \((h, k, l)\) is the center of the sphere.
• \(r\) is the radius of the sphere.

When the equation involves other terms, like linear expressions or constants, rearranging it requires completing the square for each variable.

By transforming the given example equation, we can translate it into a cleaner form that highlights its geometric properties. Understanding the standard form helps not just in this exercise but in any mathematical situation involving spheres.