Problem 55 - Thomas Calculus

Question

Find the centers and radii of the spheres in Exercises \(55-58 .\) $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

Step-by-Step Solution

Answer

The center is \((-2, 0, 2)\) and the radius is \(2\sqrt{2}\).

Step 1: Recognize the Standard Form of a Sphere

The standard form of the equation of a sphere is \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\] where \((h, k, l)\) is the center and \(r\) is the radius of the sphere.

Step 2: Rearrange the Given Equation

To identify the center and radius, rewrite the given equation in the standard form by completing the square: \[ x^{2}+y^{2}+z^{2}+4x-4z=0 \] Separate the terms related to \(x\) and \(z\): \[ (x^2 + 4x) + y^2 + (z^2 - 4z) = 0 \]

Step 3: Complete the Square for x and z

Complete the square for the \(x\) and \(z\) terms:For \(x^2 + 4x\),- Take half of the coefficient of \(x\), square it, add and subtract it: \(x^2 + 4x = (x^2 + 4x + 4 - 4) = (x + 2)^2 - 4\).For \(z^2 - 4z\),- Take half of the coefficient of \(z\), square it, add and subtract it: \(z^2 - 4z = (z^2 - 4z + 4 - 4) = (z - 2)^2 - 4\).Substituting back, we get:\[(x + 2)^2 + y^2 + (z - 2)^2 - 4 - 4 = 0\]

Step 4: Simplify the Equation

Combine and simplify the constants:\[(x + 2)^2 + y^2 + (z - 2)^2 - 8 = 0\]Add 8 to both sides to fully complete the square:\[(x + 2)^2 + y^2 + (z - 2)^2 = 8\]

Step 5: Identify the Center and the Radius

From the equation \[(x + 2)^2 + y^2 + (z - 2)^2 = 8\],we can see that the center is \((-2, 0, 2)\) and the radius is \(\sqrt{8} = 2\sqrt{2}\).

Key Concepts

Standard Form of a Sphere

The standard form of a sphere's equation provides a structured way to identify the center and radius of the sphere from the equation directly. It is expressed as:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\],where

• \((h, k, l)\) denotes the spherical center's coordinates.
• \(r\) is the sphere's radius.

Each variable's terms are squared and add up to equal the square of the radius, \(r^2\).
This form is very useful because it readily translates to the geometric representation of a sphere in three-dimensional space.
By learning to recognize the standard form, you can easily determine the sphere's center and size from any given equation.

Completing the Square

"Completing the square" is a mathematical method used to rewrite quadratic equations into a more useful form. It is often used to transform a general quadratic equation into a "perfect square trinomial." This approach is particularly helpful for equations of spheres.
Here's how the process works:

• Identify the quadratic and linear terms for each variable (e.g., identify the \(x^2 + 4x\) and the \(z^2 - 4z\) in the given equation).
• To complete the square for each variable, take half of the linear coefficient, square it, and add and subtract this value in the equation. For example:
  For \(x^2 + 4x\):
  1. Half of \(4\) is \(2\), and square \(2\) to get \(4\).
  2. Add and then subtract \(4\) within the equation: \((x^2 + 4x + 4 - 4)\) simplifies into \((x+2)^2 - 4\).
• Repeat this for other variables as needed, such as \(z\) in the equation.

Completing the square ultimately helps adjust the equation into one that matches the standard form of a sphere, revealing more about its properties.

Equation of a Sphere

A sphere's equation in three-dimensional geometry describes all points that are equidistant from a single point called the center. In the exercise, we started with the equation:\[x^2+y^2+z^2+4x-4z=0\].
Initially, this doesn't appear in the standard sphere form. Hence, we need to rewrite it so we can easily identify the sphere's center and radius.
By applying algebraic techniques like rearranging and completing the square, the equation becomes:\[(x+2)^2 + y^2 + (z-2)^2 = 8\].From here, you can see:

• The center is located at \((-2, 0, 2)\).
• The radius, \(r\), is \(\sqrt{8}\), which simplifies to \(2\sqrt{2}\).

Understanding how to read and convert an equation into this form helps visualize the sphere's characteristics and properties, enabling better problem-solving and comprehension of geometric concepts.