Problem 11 - Calculus: Early Transcendentals

Question

Find an equation of the sphere with center at (2,1,-1) and radius \(4 .\) Find an equation for the intersection of this sphere with the yz-plane; describe this intersection geometrically.

Step-by-Step Solution

Answer

The sphere's equation is \((x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 16\). It intersects the yz-plane as a circle \((y - 1)^2 + (z + 1)^2 = 12\) with radius \(2\sqrt{3}\).

Step 1: Understand Sphere Equation

The general equation for a sphere with center at \((h, k, l)\) and radius \(r\) is given by the equation:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]Given the center is \((2, 1, -1)\) and the radius is \(4\), we'll use these in the equation.

Step 2: Substitute Values into Sphere Equation

Substitute the center \((2, 1, -1)\) and radius \(4\) into the equation:\[(x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 4^2\]Expanding this, the equation becomes:\[(x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 16\]

Step 3: Sphere's Intersection with yz-plane

In the yz-plane, the x-coordinate is zero. Substitute \(x = 0\) into the sphere equation:\[(0 - 2)^2 + (y - 1)^2 + (z + 1)^2 = 16\]Calculate:\[4 + (y - 1)^2 + (z + 1)^2 = 16\]This simplifies to:\[(y - 1)^2 + (z + 1)^2 = 12\]

Step 4: Describe Geometric Intersection

The equation \((y - 1)^2 + (z + 1)^2 = 12\) represents a circle in the yz-plane. The circle has a center at \((0, 1, -1)\) with a radius of \(\sqrt{12} = 2\sqrt{3}\). This describes a cross-section of the sphere with the yz-plane.

Key Concepts

Center of a Sphere

The center of a sphere is a crucial point in geometry. It is the three-dimensional equivalent to the center of a circle in two dimensions. Understanding its location is essential when working with the equation of a sphere.

• Identified as a point \( (h, k, l) \) in space, where each coordinate represents a position along the x, y, and z axes respectively.
• The center acts as the anchor point from which every point on the sphere's surface is equidistant, known as the radius.

Finding the center of a sphere in a given equation helps you understand its position in space. In our exercise, the sphere's center is given as \( (2, 1, -1) \). It tells us that the sphere is positioned 2 units along the x-axis, 1 unit along the y-axis, and 1 unit below the xy-plane (i.e., -1 unit on the z-axis). Knowing the center provides a reference for understanding other properties, like its intersection with surfaces.

Equation of a Sphere

The equation of a sphere forms the backbone of many geometric calculations in three-dimensional space. This equation is derived using the center and radius of the sphere.

• General form: \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \).
• Each \( (x, y, z) \) represents variable points on the surface of the sphere.
• The terms \( (h, k, l) \) refer to the center of the sphere, and \( r \) represents the radius.

For the sphere featured in our problem, with center \( (2, 1, -1) \) and radius \( 4 \), the equation is:\[ (x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 16 \]This equation provides a blueprint, ensuring any point \( (x, y, z) \) lying on the sphere's surface will satisfy it. The power of this mathematical expression resides in its simplicity, giving you an efficient way to describe spherical shapes in space.

Intersection with Plane

The intersection of a sphere with a plane is a fascinating study in geometry. It combines circular and spherical geometry, revealing how curved surfaces interact with flat planes.

• A plane can slice through, merely touch, or miss a sphere entirely. Each case presents different geometrical scenarios.
• When intersecting, the result is typically a circle or a point, depending on the plane's position relative to the sphere.

In our exercise, we focus on how this sphere intersects the yz-plane, where \( x = 0 \). Substituting \( x = 0 \) in the sphere's equation simplifies it to:\[ (y - 1)^2 + (z + 1)^2 = 12 \]This new equation depicts a circle, indicating that the yz-plane slices through our sphere, forming a circular intersection. This intersection can provide insights into the sphere's spatial orientation and its relationship with other geometric shapes or objects in the vicinity.

Geometric Intersection of Spheres

Understanding the intersection of a sphere with another geometric shape, such as a plane, can reveal much about three-dimensional geometry. This concept goes beyond simple calculations to explain complex spatial relationships.

• When a sphere intersects with a plane, the result is oftentimes a circle. The shape and size of this circle depend on the position and orientation of the plane.
• The intersection circle's center is derived by projecting the center of the sphere onto the plane.

For our particular problem, the circle formed at the intersection with the yz-plane is centered at \( (0, 1, -1) \), with a radius of \( \sqrt{12} = 2\sqrt{3} \). This radius can be understood as the distance from the intersection center to any point on the circle's circumference, which exists entirely within the yz-plane. This geometrical intersection offers a cross-sectional view of the sphere, providing a tangible link between the abstract mathematical framework and real-world spatial perception.