Results 1 to 2 of 2

Math Help - Distances (in general)

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    28

    Distances (in general)

    I've asked several of my friends about this question, but nobody knows where to start with it, and unfortunately, neither do I. Here it is:

    Given point A(x1,y1) and the line ax+by+c=0, show the distance from Point A to the line is d=[|ax1+by1+c|/root(a^2 + b^2)]. Note that ax+by+c=0 corresponds to a vertical line if b=0 and to a horizontal line if a=0.

    Any help would be appreciated. Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2008
    Posts
    80
    Hello,

    It depends on what you can use as tools. (For example, what is the definition of a "distance"? Can we use vectors or the complex plane? etc.)

    The easiest way would be to use the Cauchy-Schwarz inequality: \sqrt{a^2+b^2}\sqrt{p^2+q^2}\geq |ap+bq|.
    We have to minimize (x-x_1)^2+(y-y_1)^2 where ax+by+c=0.
    By the above inequality, \sqrt{a^2+b^2}\sqrt{(x-x_1)^2+(y-y_1)^2}\geq |a(x-x_1)+b(y-y_1)|. Substitute ax+by=-c into this and you get the formula.

    Let B(x_0, y_0) be the point on the line realizing the distance: d=AB. If you know that AB is perpendicular to the line, there are many more proofs.For example, the line AB is -b(x-x_1)+a(y-y_1)=0, so you can determine B with the condition ax_0+by_0+c=0. (You can rephrase this by using vectors and saying that (x-x_1, y-y_1) is parallel to (a, b).)

    If you know trigonometrics, represent any point on the line by (x_1+r\cos t, y_1+r\sin t). As it is on the line, a(x_1+r\cos t)+b(y_1+r\sin t)+c=0 which is ax_1+by_1+c+r\sqrt{a^2+b^2}\sin (t+\theta)=0 for some \theta. The minimum of r is attained when \sin(t+\theta) is either 1 or -1.

    Bye.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. heights and distances
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: March 22nd 2011, 03:13 AM
  2. Distances
    Posted in the Geometry Forum
    Replies: 10
    Last Post: December 10th 2010, 03:29 PM
  3. Distances and velocities
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 17th 2009, 01:23 PM
  4. distances with angles
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: September 14th 2008, 02:41 PM
  5. Need help with figuring out distances
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 26th 2007, 02:56 AM

Search Tags


/mathhelpforum @mathhelpforum