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Math Help - Multivariate Calculus problem. Please help!

  1. #1
    ka_rei
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    Multivariate Calculus problem. Please help!

    Please help me with this problem, as soon as possible.

    Find the point on the curve x^2/4 + y^2 = 1 which is as close as possible to the line y = 5 -x.

    The hint on the problem says one way of minimize the distance between 2 curves is to pick generic points on both curves and the minimize the (square of ) the distant between 2 points.

    I can't think of anything at all except the fact that the shortest distance between 2 curves are always along the line that is normal to both of them.

    I have no idea where to start. Someone, Please, help me with this. Thank you!
    Last edited by ka_rei; May 1st 2007 at 05:14 AM.
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  2. #2
    TD!
    TD! is offline
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    Quote Originally Posted by ka_rei View Post
    I can't think of anything at all accept the fact that the shortest distance between 2 curves are always along the line that is normal to both of them.
    This is a good observation which allows an alternative solution (not the way where you pick two generic points, minimizing the distance).

    The given line has slope -1, so the normal line has slope 1, making it of the form y = x + k. Now you want to know the point on the ellipse where this line intersects it perpendicularly.

    Implicit differentiation yields that the tangent direct is -x/(4y). This is normal to the slope 1 if their product is -1, so: -x/(4y) = -1 <=> x = 4y. Substitution in the equation of the ellipse now gives a quadratic equation in x or y.
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