Find equations of two spheres that are centered at the origin and are tangent to the
sphere of radius r = 1 centered at (3, −2, 4).

To find the equations of two spheres that are centered at the origin and are tangent to the sphere of radius \( r = 1 \) centered at \( (3, -2, 4) \), we follow these steps:

### Step 1: Equation of a Sphere
The general equation of a sphere centered at the origin with radius \( R \) is given by:
\[
x^2 + y^2 + z^2 = R^2
\]

We need to determine the radii \( R_1 \) and \( R_2 \) of two spheres centered at the origin that are tangent to a given sphere.

### Step 2: Equation of the Given Sphere
The equation of the given sphere with radius \( 1 \) and center at \( (3, -2, 4) \) is:
\[
(x - 3)^2 + (y + 2)^2 + (z - 4)^2 = 1^2
\]
or equivalently:
\[
(x - 3)^2 + (y + 2)^2 + (z - 4)^2 = 1
\]

### Step 3: Distance Between Centers
The distance between the center of the sphere at the origin \( (0, 0, 0) \) and the center of the given sphere \( (3, -2, 4) \) is calculated using the distance formula:
\[
d = \sqrt{(3 - 0)^2 + (-2 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 4 + 16} = \sqrt{29}
\]

### Step 4: Radii of the Spheres
For the spheres to be tangent, the sum or difference of their radii must equal the distance between their centers. Therefore, we have two possible cases for the radii:

1. **Outer tangency**: The radius of the sphere centered at the origin \( R_1 \) must satisfy:
   \[
   R_1 + 1 = \sqrt{29}
   \]
   Solving for \( R_1 \):
   \[
   R_1 = \sqrt{29} - 1
   \]

2. **Inner tangency**: The radius of the sphere centered at the origin \( R_2 \) must satisfy:
   \[
   R_2 - 1 = \sqrt{29}
   \]
   Solving for \( R_2 \):
   \[
   R_2 = \sqrt{29} + 1
   \]

### Step 5: Equations of the Two Spheres
The equations of the two spheres centered at the origin are:

1. For the outer tangency:
   \[
   x^2 + y^2 + z^2 = (\sqrt{29} - 1)^2
   \]
   Simplifying:
   \[
   x^2 + y^2 + z^2 = 30 - 2\sqrt{29}
   \]

2. For the inner tangency:
   \[
   x^2 + y^2 + z^2 = (\sqrt{29} + 1)^2
   \]
   Simplifying:
   \[
   x^2 + y^2 + z^2 = 30 + 2\sqrt{29}
   \]

Thus, the equations of the two spheres are:
1. \( x^2 + y^2 + z^2 = 30 - 2\sqrt{29} \)
2. \( x^2 + y^2 + z^2 = 30 + 2\sqrt{29} \)

Do you have any questions or need further details?

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Here are 5 related questions to expand your understanding:
1. How do you find the distance between two points in 3D space?
2. How would the equations change if the spheres were not centered at the origin?
3. What is the geometric interpretation of inner and outer tangency between two spheres?
4. How can we generalize this method to spheres of different radii?
5. What would happen if the two spheres intersect instead of being tangent?

**Tip**: Always verify the center and radius of a sphere before attempting to find tangency conditions with another!

Find equations of two spheres that are centered at the origin and are tangent to the sphere of radius r = 1 centered at (3, −2, 4).

Find the equations of two spheres centered at the origin that are tangent to a sphere of radius 1 centered at (3, −2, 4). This problem involves calculating distances between sphere centers and applying the conditions for tangency. Suitable for students in Grades 10-12.

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