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Range of the parameter of sphere intersecting with a plane

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the range of the parameter d for which the intersection of the sphere x2+y2+z2=1 and the plane x+y+z=d is non-empty.


    2. Relevant equations
    Cartesian coordinates of a sphere:
    x=rcos[tex]\theta[/tex]sin[tex]\phi[/tex]
    y=rsin[tex]\theta[/tex]sin[tex]\phi[/tex]
    z=rcos[tex]\phi[/tex]


    r=1

    3. The attempt at a solution
    I substitute x,y,z in both equations
    d=sin[tex]\theta[/tex]sin[tex]\phi[/tex]+cos[tex]\phi[/tex]+cos[tex]\theta[/tex]sin[tex]\phi[/tex]
    cos2[tex]\theta[/tex]sin2[tex]\phi[/tex]+sin2[tex]\theta[/tex]sin2[tex]\phi[/tex]+cos2[tex]\phi[/tex]=1

    Since sin2[tex]\theta[/tex]+cos2[tex]\theta[/tex]=1
    I get 1+cos2[tex]\phi[/tex]=1
    This implies that [tex]\phi[/tex]=90
    Which solves the first equation for d=sin[tex]\theta[/tex]+cos[tex]\theta[/tex]
    Is this right?
    Thanks for any help.
     
  2. jcsd
  3. Oct 19, 2009 #2

    LCKurtz

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    I would think your answer would need to be in the form a ≤ d ≤ b. A simpler approach might be to observe the plane will intersect the sphere if the distance of the plane from the origin is ≤ 1. This can be easily done with vectors and no need for spherical coordinates.
     
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