Find the equation of the cylinder circumscribed
about the two spheres
(x — 2)^2+ ( y — 1)^2 + z^2 = 25, x^2 + y^2 +z^2 = 25.

To find the equation of the cylinder circumscribed about the two spheres, we begin by analyzing the given equations of the spheres.

1. **First sphere**: \((x - 2)^2 + (y - 1)^2 + z^2 = 25\)
   - Center: \((2, 1, 0)\)
   - Radius: \(5\)

2. **Second sphere**: \(x^2 + y^2 + z^2 = 25\)
   - Center: \((0, 0, 0)\)
   - Radius: \(5\)

### Strategy:
The cylinder must be circumscribed about both spheres. Since a cylinder extends infinitely in one direction, the simplest assumption is that the cylinder’s axis is along the \(z\)-axis. This assumption works because both spheres are symmetric in \(z\), and their centers lie in the \(xy\)-plane.

### Step 1: Distance between the centers of the spheres
The centers of the two spheres are:
- Center 1: \((2, 1, 0)\)
- Center 2: \((0, 0, 0)\)

The distance between the centers is:
\[
d = \sqrt{(2 - 0)^2 + (1 - 0)^2 + (0 - 0)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

### Step 2: Equation of the cylinder
A general equation of a cylinder with an axis along the \(z\)-axis is of the form:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the cylinder’s base, and \(r\) is the radius.

The cylinder must contain both spheres, so the distance between the cylinder’s center and the center of each sphere must equal the radius of the sphere (which is 5). This means the cylinder’s equation must accommodate the centers \((2, 1)\) and \((0, 0)\) in the \(xy\)-plane.

To satisfy both spheres, the cylinder should be centered midway between the centers of the two spheres in the \(xy\)-plane. The midpoint of \((2, 1)\) and \((0, 0)\) is:
\[
\left(\frac{2 + 0}{2}, \frac{1 + 0}{2}\right) = (1, 0.5)
\]

Thus, the cylinder is centered at \((1, 0.5)\) in the \(xy\)-plane.

### Step 3: Radius of the cylinder
The radius of the cylinder must be large enough to encompass the centers of both spheres. The distance from the center of the cylinder, \((1, 0.5)\), to either sphere's center can be computed. We choose the center of the first sphere, \((2, 1)\):

\[
\text{Distance} = \sqrt{(2 - 1)^2 + (1 - 0.5)^2} = \sqrt{1 + 0.25} = \sqrt{1.25}
\

Find the equation of the cylinder circumscribed about the two spheres (x — 2)^2+ ( y — 1)^2 + z^2 = 25, x^2 + y^2 +z^2 = 25.

Learn how to find the equation of a cylinder circumscribed about two spheres with a step-by-step solution. This problem involves key concepts in analytic geometry, including the equations of spheres and cylinders, the distance formula, and the midpoint theorem. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation