 # Find the equation of the sphere centered at (-4, 1, 4) with radius 3. Give an equation which describes the intersection of this sphere with the plane z = 6. 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.

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

$Radius\ of\ Sphere\ r = 3$

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.