Results 1 to 3 of 3

Math Help - shortest chord

  1. #1
    Member
    Joined
    Mar 2008
    From
    ???
    Posts
    169

    shortest chord

    What normal to the curve y = x^2 forms the shortest chord?

    This is what I did:
    I used parametric form of parabola and did stuff, put it in distance formula.. d/dt = 0 it, I'm getting a complicated polynomial ( 7th degree ) after doing that..
    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 ice_syncer View Post
    What normal to the curve y = x^2 forms the shortest chord?

    This is what I did:
    I used parametric form of parabola and did stuff, put it in distance formula.. d/dt = 0 it, I'm getting a complicated polynomial ( 7th degree ) after doing that..
    I don't believe that this problem has a neat solution. I tried it for the parabola y^2=4ax, where the general point has parametric form (at^2,2at). The normal at this point has equation y-2at = -t(x-at^2), and it meets the parabola at the point with parameter s = -t-\frac2t. So if d is the length of the chord then d^2 = (at^2-as^2)^2 + (2at-2as)^2, which works out as

    d^2 = a^2\Bigl(4t^2 + 24 + \dfrac{36}{t^2} + \dfrac{16}{t^4}\Bigr).

    Differentiate that to find that the turning point (which must be a minimum) occurs when t^6 - 9t^2-8 = 0. You can factorise that as t^6 - 9t^2-8 = (t^2+1)(t^4-t^2-8). The second factor is a quadratic in t^2, and the only positive root is t^2 = \frac12\bigl(1+\sqrt{33}). Therefore the minimum occurs when t = \sqrt{\frac12\bigl(1+\sqrt{33})} \approx 1.836377.... If you now substitute the value for t^2 into the formula for d^2, taking a=1/4, and finally take the square root to get d, then you will have the length of the shortest chord.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2008
    From
    ???
    Posts
    169
    I was hoping for some answer that uses geometry and trigonometry to find the length, rather than using distance formula
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Circle Geometry: Shortest Chord
    Posted in the Geometry Forum
    Replies: 4
    Last Post: July 7th 2010, 10:08 PM
  2. Chord and Circle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 7th 2010, 07:22 PM
  3. Need help with chord length.
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: November 5th 2009, 10:44 AM
  4. chord
    Posted in the Algebra Forum
    Replies: 5
    Last Post: October 18th 2009, 06:56 AM
  5. Right Bisector of a Chord?
    Posted in the Geometry Forum
    Replies: 2
    Last Post: October 11th 2009, 08:34 AM

Search Tags


/mathhelpforum @mathhelpforum