91Ó°ÊÓ

The center of a sphere is a crucial component when defining its equation. In this specific exercise, we are given the center at coordinates \((1, 1, 4)\). This means that the sphere is positioned in three-dimensional space with its center at the point where:

• The x-coordinate is 1
• The y-coordinate is 1
• The z-coordinate is 4

The center's coordinates directly influence the structure and position of the sphere's equation.The equation of a sphere is generally represented as: \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]where - \(h, k, l\) are the coordinates of the center, and - \(r\) is the radius. Substituting in the center coordinates from our given problem (1, 1, 4), we start forming the sphere's equation.

The radius of a sphere represents the distance from the center to any point on its surface. In our exercise, the sphere is tangent to the \(xy\)-plane.This tangency condition provides vital information about the sphere's radius.Tangent to the \(xy\)-plane means the shortest distance between the center of the sphere and the \(xy\)-plane is equal to the radius.In this case, the z-coordinate of the center determines this distance.Thus, for our sphere with the center at \((1, 1, 4)\), the radius \(r\) is the z-coordinate value, i.e., \(r = 4\). Hence, we already know the formula for calculating the sphere's radius based on the height above the \(xy\)-plane. Now, integrate this radius back into the sphere's equation, which helps in finalizing its definition.

When a sphere is tangent to a particular plane, it touches the plane at precisely one point. For this exercise, the sphere is tangent to the \(xy\)-plane.This condition simplifies our calculations considerably.

• The tangency means that the perpendicular distance between the sphere's center and the plane is exactly equal to the sphere's radius.
• This relationship is especially significant as it allows us to set the radius as the z-coordinate from the center \((1, 1, 4)\), which is \(4\).

Using this information, we can ascertain the complete sphere equation:\[(x - 1)^2 + (y - 1)^2 + (z - 4)^2 = 4^2\]This tangency ensures that only the specified radius fits perfectly in contact with the plane, comprehensively determining the sphere's geometry.