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I need help. (I must find the equation of the sphere that is tangent to 2 planes.

  1. Feb 15, 2009 #1
    1. The problem statement, all variables and given/known data
    My professor gave us 4 points A(2,3,-5) B(-2,5,1) C(9,0,4) D(3,-5,8)
    So I must find an equation of the sphere that is tangent to 2 planes. These 2 planes are
    18x + 39y - z= 158
    18 x + 39y -z = -149

    2. Relevant equations

    equation of sphere C(h,k,l) (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

    3. The attempt at a solution
    Ok How do I find it's center and it's radius. I'm not sure as to what I do. I've never seen this type of problem written like this before. So I need help. My professor gave us a hint. He said something about the D point. Will the distance formula give me the center. From A to D.

    The Vectors of these planers are <18,39, -1>
     
    Last edited: Feb 15, 2009
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  3. Feb 15, 2009 #2

    Office_Shredder

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    Notice that the planes are parallel (i.e. don't intersect) so always maintain the same distance. Find how far apart the planes are when traveling along the perpendicular to the two planes you're given, that will give you the diameter. Then the center is just halfway between the planes when traveling along a perpendicular (notice there are lots of possibilities for this)
     
  4. Feb 15, 2009 #3

    Dick

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    I'm going to guess that A, B, C and D are supposed to be points on the sphere? Where did you get the two planes from? Notice that for any two points on the sphere, like A and B, the center of the sphere must lie on the plane that perpendicularly bisects the segment AB. You can actually find three planes to intersect to find the center of the sphere. Say AB, BC and BD.
     
  5. Feb 15, 2009 #4
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    I got the 2 planes from the 1st and 2nd question.
    Find the equation of the plane that contains A,B, and C.

    Find the equation of the plane parallel to the plane you found in part a and also contains the point D.
     
  6. Feb 15, 2009 #5
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    In order for me to find the distance between these 2 planes do I use this formula.
    D= (a x1 + b y1 + c z1 + d) / square root of (a^2 + b^2 + c^2 )

    Their's an absolute value for the numerator.
    I got 307 by adding 158 with 149.
    A= 18
    B=39
    C= -5


    D= 307 / square root of (18^2 + 39^2 + (-5)^2 ) = 7.09

    Then D/2 so 7.09/2 = 3.545
     
  7. Feb 15, 2009 #6

    Dick

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    Fine, but which sphere do you want? Given two planes there are MANY spheres tangent to both of them.
     
  8. Feb 15, 2009 #7
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    So if C is 3.545 how do I find it's points
     
  9. Feb 15, 2009 #8
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    Their are many spheres. I perpindicular to point D is the sphere that I want.
     
  10. Feb 15, 2009 #9
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    So I can choose any of those 4 points.
     
  11. Feb 15, 2009 #10

    Dick

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    That's still not very clear. I'll repeat. Do you want the sphere that goes through A, B, C and D?
     
  12. Feb 15, 2009 #11
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    So the center must be (18, 39, and -1). And the point I'll choose is D (3,-5,8)

    Then I'll use the Distance formula DC= square root of (3-18)^2 + (-5-39)^2 + (8+1)^2

    =square root of 2242

    then I square it and I get r= 2242

    so the answer is

    (x-18)^2 + (y-39)^2 + (z+1)^2 =2242

    does that look correct
     
  13. Feb 15, 2009 #12
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    I wanted to choose the sphere that goes through the point D
     
  14. Feb 15, 2009 #13

    Dick

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    Hard to say, since I haven't figured out what problem you are actually trying to solve.
     
  15. Feb 15, 2009 #14
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    I had to find the equation of the sphere tangent to the 2 planes
     
  16. Feb 15, 2009 #15

    Dick

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    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    Ah, ok. So tangent to the two planes and passing through D. I would use the unit normal to the planes and the dot product. Since you know the normal is n=(18,39,-1) the unit normal is u=n/|n|. Then d=(A-D).u is the signed distance between the two planes. And the center must be located one half that distance away in the direction u. So the center is D+u*d/2.
     
  17. Feb 15, 2009 #16
    Re: I need help. (I must find the equation of the sphere that is tangent to 2 planes

    You sure this will work. I will try this out. Thanks man.
     
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