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Math Help - Find 2 spheres tangent to another sphere.

  1. #1
    Member nautica17's Avatar
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    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?
    Last edited by nautica17; January 19th 2011 at 04:26 PM. Reason: Put an x instead of a z
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  2. #2
    A Plied Mathematician
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    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?
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  3. #3
    Member nautica17's Avatar
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    o.m.g. that makes sense! I think I got it, I'll post my work in a minute.
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  4. #4
    Member nautica17's Avatar
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    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?
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  5. #5
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    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?
    Last edited by Ackbeet; January 19th 2011 at 04:35 PM. Reason: Plural for equations.
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  6. #6
    Member nautica17's Avatar
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    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?
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  7. #7
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    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*
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  8. #8
    Member nautica17's Avatar
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    Awesome! Thanks!
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  9. #9
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    So, just to be sure of things, what's your final answer?
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  10. #10
    Member nautica17's Avatar
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    Well, I'm thinking of going with the minus 1 for the first sphere and plus 1 for the second.
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  11. #11
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    I mean, take a closer look at my Post # 7. Your answer in Post # 6 is not quite correct.
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  12. #12
    Member nautica17's Avatar
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    Do I just square the right hand side then?
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  13. #13
    Member nautica17's Avatar
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    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
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  14. #14
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    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.
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  15. #15
    Member nautica17's Avatar
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    Cool. Thanks for sticking with me!
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