Question

Find the center and radius of the sphere defined by \(x^{2}-8 x+y^{2}+2 y+z^{2}+8=0\)

Short answer

The center is (4, -1, 0) and the radius is 3.

Step-by-step solution

Group the terms by variable

Start by grouping the terms with the same variable in the given equation: \(x^2 - 8x + y^2 + 2y + z^2 + 8 = 0\). This helps simplify the process of completing the square.

Complete the square for the x terms

For the terms involving \(x\), complete the square on \(x^2 - 8x\). Rewrite it as \((x-4)^2 - 16\). The perfect square \((x-4)^2\) requires subtracting \(16\) which will be added as a separate constant.

Complete the square for the y terms

For the terms involving \(y\), complete the square on \(y^2 + 2y\). Rewrite it as \((y+1)^2 - 1\). The perfect square \((y+1)^2\) requires subtracting \(1\).

Write z terms as a perfect square

For the \(z\) term \(z^2\), it is already a perfect square, so it remains as \((z-0)^2\).

Rewrite the equation with the completed squares

Substitute the completed squares into the original equation. It becomes: \((x-4)^2 - 16 + (y+1)^2 - 1 + (z-0)^2 + 8 = 0\).

Simplify the equation

Combine the constants \(-16 - 1 + 8 = -9\). This simplifies the equation to \((x-4)^2 + (y+1)^2 + (z-0)^2 = 9\).

Extract the center and the radius

The equation \((x-4)^2 + (y+1)^2 + (z-0)^2 = 9\) is now in standard form for a sphere, \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). Here, the center of the sphere is \((4, -1, 0)\) and the radius is \(r = \sqrt{9} = 3\).

Key concepts

Completing the Square

Completing the square is a method used to make quadratic expressions easier to work with by turning them into perfect square trinomials. This is particularly helpful in geometric problems involving circles and spheres.
Here's how it works:

• First, identify the quadratic expression for each variable.
• For each expression, rewrite it as a perfect square. This involves finding a number that makes the expression fit the pattern: \((v + b)^2 = v^2 + 2bv + b^2\)
• Add and subtract the same value to preserve the equation's equality. This added value is \( \frac{b}{2}^2\).

Let's apply this to an example: For the term \(x^2 - 8x\), find \(b\) by taking half of \(-8\), so \(-4\). Then, \((-4)^2 = 16\), so it transforms into \((x - 4)^2 - 16\).
This makes equations easier to solve for centers and radii of spheres.

Center of a Sphere

The center of a sphere is a point that is equidistant from all points on the surface of the sphere. To find the center when given a sphere equation, you convert the equation into a standard form \( (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \).
Here's how it's done:

• Complete the square for each variable term as shown previously.
• Identify the constant terms \((a, b, c)\) from each squared term after transformation.
• These constants represent the coordinates of the center.

For example, in the equation \( (x-4)^2 + (y+1)^2 + (z-0)^2 = 9 \), the center of the sphere is \( (4, -1, 0) \).
Each coordinate is the point around which each variable's squared term pivots.

Radius of a Sphere

The radius of a sphere is the constant distance from its center to any point on its surface. When an equation is in the standard form, you can find the radius by isolating the right-hand side of the equation.
It's given as \( r \) in the sphere's equation \( (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 \).
Steps to find the radius:

• Ensure the equation is in standard form by completing the square.
• The constant on the equation's right side is \( r^2 \).
• Take the square root of that constant to determine the radius.

In our example, the equation simplifies to \( r^2 = 9 \), so the radius \( r \) is \( \sqrt{9} = 3 \).
Knowing the radius helps visualize the full three-dimensional space the sphere occupies.