Question

Find the standard equation of the sphere. Center: \((-2,1,1) ;\) tangent to the \(x y\) -plane

Short answer

Therefore, the standard equation of the sphere is \((x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 1\).

Step-by-step solution

Find the radius of the sphere

Since the sphere is tangent to the xy-plane, the radius of the sphere would be equal to the absolute value of the z-coordinate of the center, hence r = |1| = 1.

Substitute the center and radius in the standard form equation

Now we implement the values of the center coordinates (-2,1,1) and the radius 1 into the standard form of a sphere's equation to get \((x - (-2))^2 + (y - 1)^2 + (z - 1)^2 = 1^2\).

Simplify the equation

After substituting, simplify the equation which yields \((x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 1\).

Key concepts

Sphere Geometry

When we think about spheres, we often imagine a perfectly round 3D object, like a basketball or Earth. In mathematics, a sphere's definition is specific: it's the set of all points in three-dimensional space that are the same distance (known as the radius) from a fixed point (the center).

To understand a sphere's surface, we often look at its intersection with planes. A plane like the xy-plane can intersect with a sphere in different ways, for example, it can pass through the sphere, touching it at one point (tangency), or it could also cut through the sphere, forming a circle.

The unique properties of a sphere are crucial when tackling problems related to its geometry. A core concept to remember is that the distance from the center of the sphere to any point on its surface is constant, defining the radius of the sphere. Furthermore, the standard equation of a sphere in three dimensions, \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \) where \( (h, k, l) \) is the center and \( r \) is the radius, helps in visualizing all points that make up this 3D shape.

Coordinate Geometry

The language of coordinate geometry allows us to translate shapes into algebraic expressions. In essence, we use a coordinate system, like the Cartesian coordinate system, to describe the location of points in space with coordinates. For a three-dimensional space, these are typically presented as \( (x, y, z) \).

Knowing this, we can define the position of objects, such as spheres, in this 3D space using equations. The standard equation of a sphere, as mentioned before, is a great example of this translation. Moreover, it's with these coordinates that we find the sphere's central point and thus, its radius, by relating it to other planes or axes.

To solve problems in coordinate geometry, it is essential to be comfortable with plotting points and understanding how the distance formula applies to points in a 3D space. The ability to visualize how 3D shapes are situated and how they interact with other geometric entities, like planes, is crucial in fully grasping the beauty of coordinate geometry.

3D Geometric Shapes

Three-dimensional shapes, or solids, encompass a wide range of figures, such as spheres, cubes, pyramids, and cylinders. Each of these shapes has unique properties that define their appearance and equations in a 3D space.

Understanding these properties allows one to calculate volumes, surface areas, centroids, and various other attributes associated with the shape. 3D geometric shapes are vital in numerous fields, from engineering to graphics design. As students progress through geometry, they will encounter problems that require them to apply their knowledge of these shapes in contexts and coordinate systems that simulate real world scenarios.

In the context of spheres, as we've seen, knowing that the sphere is defined by its center and radius enables us to derive other aspects of its geometry, such as its volume \( (\frac{4}{3}\pi r^3) \) or its surface area \( (4\pi r^2) \) with clarity and ease. As such, an understanding of 3D geometric shapes is foundational for students pursuing studies in fields where spatial reasoning is imperative.