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

Step 1: Equation of the Given Sphere

The equation of a sphere is given by: $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$ For the given sphere centered at $(3, -2, 4)$ with radius $r = 1$, the equation is: $(x - 3)^2 + (y + 2)^2 + (z - 4)^2 = 1^2$ $(x - 3)^2 + (y + 2)^2 + (z - 4)^2 = 1$

Step 2: Condition for Tangency

For the two spheres centered at the origin to be tangent to the given sphere, their distance from the origin to the center of the given sphere (which is $(3, -2, 4)$) must equal the sum of their radii. The distance between the origin $(0, 0, 0)$ and $(3, -2, 4)$ is: $d = \sqrt{(3-0)^2 + (-2-0)^2 + (4-0)^2} = \sqrt{3^2 + (-2)^2 + 4^2} = \sqrt{9 + 4 + 16} = \sqrt{29}$

Let $R$ be the radius of the sphere centered at the origin. The distance condition for tangency is: $R + 1 = \sqrt{29}$ Solving for $R$: $R = \sqrt{29} - 1$

Step 3: The First Sphere Equation (Tangency on the Outside)

The equation of the first sphere (with radius $R = \sqrt{29} - 1$) centered at the origin is: $x^2 + y^2 + z^2 = (\sqrt{29} - 1)^2 = 29 - 2\sqrt{29} + 1 = 30 - 2\sqrt{29}$ Thus, the equation of the first sphere is: $x^2 + y^2 + z^2 = 30 - 2\sqrt{29}$

Step 4: The Second Sphere Equation (Tangency on the Inside)

For tangency on the inside, the radius of the sphere centered at the origin would be $R$, such that: $R = \sqrt{29} + 1$ The equation of the second sphere (with radius $R = \sqrt{29} + 1$) is: $x^2 + y^2 + z^2 = (\sqrt{29} + 1)^2 = 29 + 2\sqrt{29} + 1 = 30 + 2\sqrt{29}$ Thus, the equation of the second sphere is: $x^2 + y^2 + z^2 = 30 + 2\sqrt{29}$

Final Answer:

• The equation of the first sphere is:
  $x^2 + y^2 + z^2 = 30 - 2\sqrt{29}$

• The equation of the second sphere is:
  $x^2 + y^2 + z^2 = 30 + 2\sqrt{29}$

Do you want more details on any step or have any questions?

Here are 5 related questions to expand your understanding:

1. How do we derive the general formula for the distance between two points in 3D space?
2. What is the geometrical significance of tangency between two spheres?
3. How can we find the point of tangency between two spheres?
4. How does the radius of a sphere affect its equation?
5. Can two spheres centered at different points have multiple tangencies?

Tip: The distance formula in 3D is a direct extension of the Pythagorean theorem into three dimensions. Always ensure you square the differences between each coordinate component!