Thread: Find 2 spheres tangent to another sphere.

1. Find 2 spheres tangent to another sphere.

This is a problem from my calculus 3 class.

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

So for the sphere they give, I have figured out the equation (or at least I hope I did), which is:
(x-3)^2 + (y+2)^2 + (z-4)^2 = 1

I figure the two spheres I need to find are going to be tangent to the given sphere internally and externally, meaning they will overlap. My question however is what direction I go about now to solve the radii of the two spheres?

I'm working on the problem as we speak, maybe I just need a light bulb moment?

2. The only piece of information you need is the radii of the two spheres. Think about a straight line going from the origin (the center of both sphere you have to find) and going straight to the center of the radius 1 sphere. What length does it have? (You'll need to compute that.) Then one sphere you're after has radius of that distance minus one. The other has radius of that distance plus one. Does that make sense?

3. o.m.g. that makes sense! I think I got it, I'll post my work in a minute.

4. sqrt(29) plus or minus 1? I just did the distance formula for the given sphere and then plus or minus 1. Is that right?

5. I would agree that those are the radii, and that you're almost done, but you need to write the equations for the spheres as your final answer. So what do you get?

6. Sphere 1 = x^2 + y^2 + z^2 = sqrt(29) - 1
Sphere 2 = x^2 + y^2 + z^2 = sqrt(29) + 1

It doesn't matter which one is plus or minus does it?

7. No, it doesn't matter which sphere is which. However, it does matter if you appropriately square the radii on the RHS or not. *ahem*

8. Awesome! Thanks!

9. So, just to be sure of things, what's your final answer?

10. Well, I'm thinking of going with the minus 1 for the first sphere and plus 1 for the second.

11. I mean, take a closer look at my Post # 7. Your answer in Post # 6 is not quite correct.

12. Do I just square the right hand side then?

13. Sphere 1 = x^2 + y^2 + z^2 = 30 - 2sqrt(29)
Sphere 2 = x^2 + y^2 + z^2 = 30 + 2sqrt(29)

I think I got it now! :P

14. Right; well, the equation for a sphere is

$(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2},$ not

$(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r,$ right?

[EDIT]: Your post # 13 is correct.

15. Cool. Thanks for sticking with me!

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find the equation of two spheres that are centered at the origin

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