Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Find the standard equation of the sphere. Center: \((1,2,0) ;\) tangent to the \(y z\) -plane

Step-by-Step Solution

Answer

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

Step 1: Identify the Center

The center of the sphere \((a, b, c)\) is given as \((1, 2, 0)\). So \(a = 1\), \(b = 2\), and \(c = 0\).

Step 2: Find the Radius

Since the sphere is tangent to the yz-plane, the radius is the distance from the center to the yz-plane. The distance from a point \((a, b, c)\) to the yz-plane is \(a\). So the radius \(r\) is equal to 1.

Step 3: Write the Equation of the Sphere

Using the formula for the equation of a sphere: \( (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \), substituting \(a = 1\), \(b = 2\), \(c = 0\) and \(r = 1\), the equation becomes: \( (x-1)^2 + (y-2)^2 + (z-0)^2 = 1^2 \). This simplifies to \( (x-1)^2 + (y-2)^2 + z^2 = 1 \).

Key Concepts

Standard Equation of a Sphere

Understanding the standard equation of a sphere is crucial for solving problems in coordinate geometry. It represents a set of all points in three-dimensional space that are at a given distance, known as the radius, from a fixed point called the center. The general form of this equation is \( (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \) where \(a, b,\, \text{and}\, c\) are the coordinates of the center, and \(r\) is the radius of the sphere.

When we have a sphere that is tangent to one of the coordinate planes, as in the given exercise, one of the coordinates of the center will be equal to the radius. For instance, if a sphere is tangent to the yz-plane, then the x-coordinate of the center is equal to the radius of the sphere. The beauty of the standard equation lies in its simplicity, and once the center and the radius are known, you can sketch the sphere and identify all the points that lie on its surface.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of a coordinate system. This branch of mathematics enables us to describe the location of points, lines, and shapes in a two-dimensional plane or three-dimensional space using coordinates.

In the context of spheres, we use a three-dimensional coordinate system to define the location of the center and calculate the radius. The coordinates are typically written in the form \( (x, y, z) \) and represent the position of a point along the x-axis, y-axis, and z-axis, respectively. Understanding the basics of coordinate geometry is essential for interpreting and solving problems related to the equations of geometric shapes, such as spheres, cylinders, or planes.

Radius of a Sphere

The radius of a sphere is perhaps one of the most intuitive yet significant concepts in geometry. It's the straight-line distance from the center of the sphere to any point on its surface. All points on the sphere's surface are equidistant from its center—this distance is the radius.

Determining the radius is a critical step in solving the standard equation of a sphere. In our exercise, since the sphere is tangent to the yz-plane at a point, and we know the center, the x-coordinate of the center directly gives us the radius, which is the shortest distance to the tangent plane. This simplifies our calculations and provides a concrete dimension that characterizes the size of the sphere. Whether comparing volumes or surface areas, the radius is fundamental, and all other measurements of the sphere derive from it.