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Math Help - A point is constrained to a line and a plane. Find closest and farthest points from O

  1. #1
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    Lagrange Multiplier help. Find max/min distance from origin.

    I have a problem that I need help with. I know I have to use Lagrange multipliers for this question but I am confused about what equation I want to maximize and minimize... Any help is appreciated.

    A point is constrained to lie on the surface x2+ y2=1 and the plane 2x+3y+z=1 What are the closest and farthest points from the origin and where do they occur?

    I used Lagrange Multipliers for the solution.

    I assumed that I wanted to minimize/maximize f(x,y,z)=x2+y2+z2 ... I'm not sure if this is right.
    Then I set up the lagrange equation L(x,y,z,lamda1,lamda2)
    Then I differentiated wrt all variables and set the equations = 0.
    I solved for the values of lamda1 and lamda2 then for x, y and z.

    I concluded that the points occur at f(+-2/sqrt(13),+-3/sqrt(13),0)... Originally I thought this was correct because I though of x2+ y2=1 as a circle but I now realize it is a cylinder... Please can someone check this out!
    Last edited by lytwynk; March 11th 2013 at 10:33 PM.
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  2. #2
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    Re: A point is constrained to a line and a plane. Find closest and farthest points fr

    The values for x and y are correct, but note that if x is positive then so is y, and if x is negative then so is y.
    Your value for z is wrong though, it doesn't satisfy the second constraint, from which its values, (for each pair of x and y), can be calculated.
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  3. #3
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    Re: A point is constrained to a line and a plane. Find closest and farthest points fr

    Ok... thanks. Also, when solving the equations there was also another point to test where z actually does equal zero. In this case I would just have to solve x^2+y^2=1 and also 2x+3y=1... is this also right?
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