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Math Help - The points closest to the origin

  1. #1
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    The points closest to the origin

    Question: The intersection of the plane x+2y+3z=10 and the surface z=x^2+y^2 is an ellipse E. Find the points on E that are closest to the origin.

    What I did:

    First I join the two equations together to form: x+2y+3x^2+3y^2=10

    Using d^2=x^2+y^2+z^2 and treating the above equation as a constraint:
    2x=(1+6x)\lambda
    2y=(2+6y)\lambda
    x+2y+3x^2+3y^2=10

    I managed to get: (x=2/3,y=4/3),(x=-1,y=-2)

    Subst back into the equation I have z= 20/9 \text{ and } 5 respectively.

    So the points I conclude is (2/3,4/3,20/9)

    Is this right? Or do I have to form two contraint equations instead?
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  2. #2
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    What you have done is perfectly valid.
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