Question

Complete the squares to find the center and radius of the sphere whose equation is given (see Example 2). \(4 x^{2}+4 y^{2}+4 z^{2}-4 x+8 y+16 z-13=0\)

Short answer

Center: \( (\frac{1}{2}, -1, -2) \); Radius: \( \sqrt{\frac{17}{2}} \)."

Step-by-step solution

Rewrite the Equation

Rewrite the given equation by grouping the terms for each variable. The equation is: \(4(x^2 - x) + 4(y^2 + 2y) + 4(z^2 + 4z) = 13\).

Factor Out the Common Factor

Factor out the common factor of 4 from each grouped term: \(4(x^2 - x) + 4(y^2 + 2y) + 4(z^2 + 4z) = 13\). This becomes \(4[(x^2 - x) + (y^2 + 2y) + (z^2 + 4z)] = 13\).

Complete the Square for x

Complete the square for the expression \(x^2 - x\). To do this, take half of the coefficient of \(x\), square it, and add/subtract inside the parentheses: \( x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4} \).

Complete the Square for y

Complete the square for \(y^2 + 2y\). Half of 2 is 1, so: \( y^2 + 2y = (y + 1)^2 - 1 \).

Complete the Square for z

Complete the square for \(z^2 + 4z\). Half of 4 is 2, so: \( z^2 + 4z = (z + 2)^2 - 4 \).

Substitute Completed Squares

Substitute back into the equation: \[ 4[(x - \frac{1}{2})^2 - \frac{1}{4} + (y + 1)^2 - 1 + (z + 2)^2 - 4] = 13 \].

Simplify

Expand the equation: \[ 4\left((x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 - \frac{1}{4} - 1 - 4\right) = 13 \]. Simplifying gives \[ 4((x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2) - 21 = 13 \].

Solve for the Sphere Equation

Add 21 to both sides to solve for the sphere equation: \[ 4((x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2) = 34 \]. Dividing through by 4 gives the sphere equation: \( (x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 = \frac{34}{4} \).

Write the Standard Form

Write the standard form to identify the center and radius: \( (x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 = \frac{17}{2} \). The center is \( (\frac{1}{2}, -1, -2) \) and the radius is \( \sqrt{\frac{17}{2}} \).

Key concepts

Equation of a Sphere

The equation of a sphere in a three-dimensional space is quite similar to the equation of a circle in two dimensions, but with an added dimension for the third axis. This equation helps in defining a perfect sphere's size and position in 3D space. The general form of the equation of a sphere is:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]Here,

• \((h, k, l)\) represents the center coordinates of the sphere,
• and \(r\) represents the radius of the sphere.

Rewriting a given complex equation into this standard form helps in easily identifying the sphere's characteristics such as its center and radius. In our original exercise, through steps involving completing the square, the original equation of the sphere was reorganized to match this standard equation form. This allows us to extract useful information about the sphere's geometry.

Center of a Sphere

The center of a sphere is a critical point that defines its position in 3D space. In the standard equation of a sphere \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\], its center is indicated by the coordinates \((h, k, l)\). With these coordinates, you can identify where exactly in the three-dimensional space the sphere is located.In our given exercise, the process of completing the square for each variable term transformed the original equation to reveal the center of the sphere:\[(x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 = \frac{17}{2}\]From this equation, the center of the sphere is identified as \((\frac{1}{2}, -1, -2)\). Each of these values is derived from the transformation of the original equation into its standard form by isolating the squared terms and identifying the constants within the parentheses.

Radius of a Sphere

The radius of a sphere measures the distance from its center to any point on its surface. It's a crucial part of the sphere's defining equation. In the standard form of the sphere's equation \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\], \(r\) is the radius.To find the radius from a complete sphere equation, solve for \(r^2\). In the exercise provided, the completed and simplified equation was:\[(x - \frac{1}{2})^2 + (y + 1)^2 + (z + 2)^2 = \frac{17}{2}\]From this equation, \(r^2 = \frac{17}{2}\), leading to the conclusion that the radius \(r\) is given by \(\sqrt{\frac{17}{2}}\). This step involves taking the square root of the constant on the right-hand side of the equation to determine the actual radius length, offering insight into the size of the sphere in space. Understanding this concept is key to apprehending how spheres occupy space and represent dimensions in three-dimensional geometry.