Problem 56 - Thomas Calculus

Question

Find the centers and radii of the spheres in Exercises \(55-58 .\) $$x^{2}+y^{2}+z^{2}-6 y+8 z=0$$

Step-by-Step Solution

Answer

Center: (0, 3, -4); Radius: 5

Step 1: Identify the Sphere Equation

The given equation is \( x^2 + y^2 + z^2 - 6y + 8z = 0 \). This is not in the standard form of a sphere equation, which is \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). To find the center \((h, k, l)\) and radius \(r\), we need to rearrange the given equation into the standard form by completing the square.

Step 2: Rearrange and Complete the Square

Hold the terms involving \(x\), \(y\), and \(z\) separately: \( x^2 + (y^2 - 6y) + (z^2 + 8z) = 0 \). Complete the square for the \(y\) terms and \(z\) terms.

Step 3: Completing the Square for y

For \(y^2 - 6y\): Take half of \(-6\), which is \(-3\), and square it to get \(9\). Add and subtract \(9\) inside the parentheses: \( (y^2 - 6y + 9 - 9) \), thus \( (y - 3)^2 - 9 \).

Step 4: Completing the Square for z

For \(z^2 + 8z\): Take half of \(8\), which is \(4\), and square it to get \(16\). Add and subtract \(16\) inside the parentheses: \( (z^2 + 8z + 16 - 16) \), thus \( (z + 4)^2 - 16 \).

Step 5: Write in Standard Form

Substituting back the completed squares into the equation gives: \((x^2 + (y - 3)^2 - 9 + (z + 4)^2 - 16) = 0 \). Simplify by collecting constants to get \(x^2 + (y - 3)^2 + (z + 4)^2 = 25 \).

Step 6: Identify the Center and Radius

This equation \( x^2 + (y - 3)^2 + (z + 4)^2 = 25 \) now matches the standard form \( (x - 0)^2 + (y - 3)^2 + (z + 4)^2 = 5^2 \). Therefore, the center is \((0, 3, -4)\) and the radius \( r \) is \(5\).

Key Concepts

Center of a Sphere

When given the equation of a sphere, the first step to finding the center is to rearrange it into the standard form, \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). The center of the sphere is represented by the coordinates \((h, k, l)\). In this form,

• \(h\) is the x-coordinate of the center,
• \(k\) is the y-coordinate of the center,
• \(l\) is the z-coordinate of the center.

The given equation in our exercise was \(x^2 + y^2 + z^2 - 6y + 8z = 0\), which is not in standard form yet. By completing the square, we identify the coordinates of the center after transformation: \((0, 3, -4)\). This means the sphere is centered 3 units away along the y-axis and 4 units away along the z-axis in the negative direction from the origin. This rearrangement from the expanded to standard form helps visualize and understand the sphere's position in 3D space.

Radius of a Sphere

The radius of a sphere can be easily identified once the equation of a sphere is written in the standard form, \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). Here, \(r\) represents the radius of the sphere. The question becomes simple once we know the value of \(r^2\) from the equation. In the exercise, by rearranging terms and completing the square, we found the equation \(x^2 + (y - 3)^2 + (z + 4)^2 = 25\). This equivalent equation is in the standard form where the radius squared, \(r^2\), is equal to 25.

• Taking the square root of both sides, the radius \(r\) becomes 5.
• This means, the length from the center \((0, 3, -4)\) to any point on the surface of the sphere is 5 units.

Determining the radius in this way provides a clear visual of the sphere's size relative to its center.

Completing the Square

Completing the square is an essential algebraic technique used to convert quadratic expressions into a perfect square trinomial. This technique is particularly useful for rewriting a quadratic equation in standard form. In the sphere equation, the goal is to isolate terms involving a single variable and convert them into the form \((variable - constant)^2\).
To complete the square, follow these steps for each variable:

• Focus on the terms involving one variable, such as \(y: y^2 - 6y\).
• Take half the coefficient of the linear term, here it is -6, and square it to get 9.
• Add and subtract this squared term within the equation: \(y^2 - 6y + 9 - 9\), simplifying to \((y - 3)^2 - 9\).

Similarly, repeat for the \(z\) terms:

• For \(z: z^2 + 8z\), half of 8 is 4, squared is 16.
• Adjust the expression: \(z^2 + 8z + 16 - 16\) becomes \((z + 4)^2 - 16\).

Combining these steps ensures that we successfully transform the sphere's equation into its standard form. This approach not only aids in determining the center and radius but also enhances the understanding of the geometric properties of spheres.