Results 1 to 9 of 9

Math Help - Point on Circle Closest to Another Point on Circle

  1. #1
    Junior Member
    Joined
    Sep 2011
    Posts
    29

    Point on Circle Closest to Another Point on Circle

    Hi all ---



    I understand the circle equations. And I understand the vector joining the two centers of the circles is just ---
     [1,1] - [5, 4] = [-4, -3] so its norm is \sqrt{(-4)^2 + (-3)^2}

    But I don't understand the \frac{2}{5}(-4, -3) in the last line at all. Where does it come from? What does it mean?

    Thanks a lot ---
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,888
    Thanks
    683

    Re: Point on Circle Closest to Another Point on Circle

    distance between the two centers is 5

    radius of the first circle is 2

    point on the first circle closest to the second lies 2/5 of the total distance from center to center.

    start position (center of the first circle) + 2/5 of the direction vector from center to center

    (5,4) + (2/5)<-4,-3>
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2011
    Posts
    29

    Re: Point on Circle Closest to Another Point on Circle

    Quote Originally Posted by skeeter View Post
    distance between the two centers is 5

    radius of the first circle is 2

    point on the first circle closest to the second lies 2/5 of the total distance from center to center.
    Hi skeeter ---

    Thanks for answering. The part above I get. But the rest of your post I don't, sadly.

    I drew a sketch ---





    Because [5,4] = vector to the center of the large circle is NOT on the same line as [4, 3] = distance between the two centers ---

    how does [ 5, 4] - (2/5)[4, 3] give us the point we want?

    I get your post if the green and red lines are collinear, but that's not the case here?

    Thanks a lot ---

    start position (center of the first circle) + 2/5 of the direction vector from center to center

    (5,4) + (2/5)<-4,-3>
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,888
    Thanks
    683

    Re: Point on Circle Closest to Another Point on Circle

    think of it as adding two position vectors ...

    (5i + 4j) + \frac{2}{5}(-4i - 3j)

    the sum of the two vectors is the position on the large circle closest to the center of the small circle.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2011
    Posts
    29

    Re: Point on Circle Closest to Another Point on Circle

    Hi skeeter ---

    Thanks so much for answering.

    I get how to do arithmetic with vectors, but I still somehow don't understand why [5,4] - (2/5)[4, 3] works, probably because [4, 3] and [5,4] are NOT collinear.

    Here's what I'm thinking --- (2/5)[4, 3] gives the radius of the big circle right on [4, 3] - the vector joining the two circles's centers.

    But doing [5,4] - (2/5)[4, 3] looks to me like we're getting the part/rest of the red vector that's outside the big circle. But the red vector isn't on [4, 3] - so it wouldn't give the point on the big circle closest to the small circle?

    And I know that the part of the green and red vectors inside the big circle have the same length in the big circle, because they are both radii.
    Last edited by mathminor827; September 19th 2011 at 12:50 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,801
    Thanks
    1691
    Awards
    1

    Re: Point on Circle Closest to Another Point on Circle

    Quote Originally Posted by mathminor827 View Post
    I get how to do arithmetic with vectors, but I still somehow don't understand why [5,4] - (2/5)[4, 3] works, probably because [4, 3] and [5,4] are NOT collinear.
    The set of points (5-t,4-3t)~0\le t\le 1) is the line segment between the centers.
    When t=0 we get (5,4).
    When t=1 we get (4,1).
    When t=\frac{2}{5} we get the point we want.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Sep 2011
    Posts
    29

    Re: Point on Circle Closest to Another Point on Circle

    Quote Originally Posted by Plato View Post
    The set of points (5-t,4-3t)~0\le t\le 1) is the line segment between the centers.
    When t=0 we get (5,4).
    When t=1 we get (4,1).
    When t=\frac{2}{5} we get the point we want.
    Hi Plato ---

    Thanks for answering. But aren't you working backwards from the answer? I mean - I understand what you're saying.

    But I'm stuck on the idea or process to actually get the answer - I'm not asking about the actual values of the coordinate.

    Thanks ---
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,801
    Thanks
    1691
    Awards
    1

    Re: Point on Circle Closest to Another Point on Circle

    Quote Originally Posted by mathminor827 View Post
    But aren't you working backwards from the answer? I mean - I understand what you're saying.
    Absolutely not!
    The length of that line segment between centers is 5.
    The radius of the circle centered at (5,4) is 2.
    From (5,4) we want a point that is \frac{2}{5} the way between the centers.
    What is backwards about that?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    914
    Thanks
    27

    Re: Point on Circle Closest to Another Point on Circle

    centers of circles 5,4 and 1,1 give a slope diagram delta y =3 delta x=4. hypo =5. the point required is three units from 1,1 along the hypo. By similar triangles 3/5=h/4 h=12/5 x= 1+12/5 =17/5 =3.4
    3/5=v/3 v=9/5 y= 9/5 +1 =14/5 = 2.8
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 6th 2011, 06:44 AM
  2. Circle, tangent line, and a point not on the circle
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: March 31st 2011, 01:40 PM
  3. Finding closest point on line to a point
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 3rd 2010, 11:28 AM
  4. Replies: 2
    Last Post: November 5th 2009, 12:05 PM
  5. Replies: 1
    Last Post: April 27th 2009, 02:30 PM

/mathhelpforum @mathhelpforum