This question aims to find the equation of the **sphere centered** at **(-4, 1, 4)** in **3D coordinates** and also an equation to describe the **intersection** of this **sphere** with a **plane z=6. **

The question is based on the concepts of a **solid geometry. Solid geometry** is the part of mathematics **geometry** that deals with **solid shapes** like **spheres, cubes, cylinders, cones,** etc. These shapes are all represented in **3D coordinate systems.**

## Expert Answer

The given information about this question is as follows:

\[ Center\ of\ Sphere\ c = ( -4, 1, 4) \]

\[ Radius\ of\ Sphere\ r = 3 \]

The **general equation** for any **sphere** with **center** $c = (x_0, y_0, z_0)$ and **radius** **r** is given as:

\[ ( x\ -\ x_0 )^2 + ( y\ -\ y_0 )^2 + ( z\ -\ z_0 )^2 = r^2 \]

Substituting the values of this **sphere** in the **general equation,** we get:

\[ ( x\ -\ (-4))^2 + ( y\ -\ 1 )^2 + (z\ -\ 4 )^2 = 3^2 \]

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( z\ -\ 4)^2 = 9 \]

This equation represents the **sphere,** which has a **radius** of **3**, and it is **centered** at **c = (-4, 1, 4).**

To find the equation of the **intersection** of the **plane** of this **sphere,** we simply need to put the value of **z,** which is a **plane** in the equation of the **sphere.** Substituting the value of **z** in the above equation, we get:

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( 6\ -\ 4)^2 = 9 \]

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( 2 )^2 = 9 \]

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + 4 = 9 \]

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 9\ -\ 4 \]

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 5 \]

This represents the **intersection** of the **plane** with the **sphere.**

## Numerical Result

The **equation** of the **sphere** is calculated to be:

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( z\ -\ 4)^2 = 9 \]

The **equation** representing the **intersection** of the **sphere** with the **plane** **z=6** is calculated to be:

\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 5 \]

## Example

Find the equation of the sphere **centered** at **(1, 1, 1)** and **radius** equal to **5**.

\[ Center\ of\ Sphere\ c = ( 1, 1, 1) \]

\[ Radius\ of\ Sphere\ r = 5 \]

Using the **general equation** of the **sphere,** we can calculate the equation of the **sphere** with **radius** **5 centered** at** (1, 1, 1).**

\[ ( x\ -\ x_0 )^2 + ( y\ -\ y_0 )^2 + ( z\ -\ z_0 )^2 = r^2 \]

Substituting the values, we get:

\[ ( x\ -\ 1 )^2 + ( y\ -\ 1 )^2 + ( z\ -\ 1 )^2 = 5^2 \]

\[ ( x\ -\ 1 )^2 + ( y\ -\ 1 )^2 + ( z\ -\ 1 )^2 = 25 \]

This is the equation of the **sphere centered** at **(1, 1, 1)** with a **radius** of **5 units.**