Results 1 to 3 of 3

Math Help - Find the minimum distance between the curves

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    8

    Find the minimum distance between the curves

    Find the minimum distance between the curves y^2 = x-1 and x^2 = y-1
    Please tell the procedure, how to solve this?


    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by utsav View Post
    Find the minimum distance between the curves y^2 = x-1 and x^2 = y-1
    Please tell the procedure, how to solve this?
    For this particular problem, if you draw a diagram you'll see that the two parabolas are symmetric about the line y=x. So the shortest distance between them will be between the two points where the tangents are parallel to that line.

    The points on the parabolas where the tangents have gradient 1 are \bigl(\tfrac12,\tfrac54\bigr) on x^2=y-1, and \bigl(\tfrac54,\tfrac12\bigr) on y^2=x-1. The distance between these points is \boxed{\tfrac34\sqrt2}.

    In general, if you can't use that sort of symmetry to simplify the problem, you would have to find the equation of the normal at a general point on one curve. Then find the point(s), if any, where that line meets the other curve, determine the distance along the normal to the closest point on the other curve, and finally minimise that distance.

    Edit. For the general case, Mr F's method (see next comment) is much better than my laborious procedure of using normals.
    Last edited by Opalg; September 13th 2009 at 04:01 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by utsav View Post
    Find the minimum distance between the curves y^2 = x-1 and x^2 = y-1
    Please tell the procedure, how to solve this?


    Thanks
    The problem can be done using calculus. The start would be to let (a, a^2 + 1) be a point on x^2 = y-1 and (b^2 + 1, b) be a point on y^2 = x-1. Substitute these points into the formula for the distance between two points to get an expression for the distance D in terms of a and b. You require the values of a and b so that the distance is a minimum.

    However ..... here's a hint for an easier way: note that the two curves are inverses of each other and are therefore reflections of each other in the line y = x.


    Edit: I was very slow. Beaten by Opalg.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Polar curves Maximum and minimum.
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: April 6th 2011, 04:39 PM
  2. Minimum distance
    Posted in the Calculus Forum
    Replies: 11
    Last Post: January 11th 2010, 05:12 AM
  3. minimum distance
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 21st 2009, 04:07 PM
  4. Find the minimum distance (Calculus II)
    Posted in the Math Challenge Problems Forum
    Replies: 3
    Last Post: April 20th 2009, 01:30 PM
  5. Minimum Distance
    Posted in the Geometry Forum
    Replies: 6
    Last Post: October 25th 2007, 02:58 AM

Search Tags


/mathhelpforum @mathhelpforum