Results 1 to 8 of 8

Math Help - includes circle,chord and tangent concepts

  1. #1
    Member grgrsanjay's Avatar
    Joined
    May 2010
    From
    chennai,tamil nadu
    Posts
    143
    Thanks
    1

    includes circle,chord and tangent concepts

    prove that chord of contact of the pair of tangents to the circle x^2 + y^2 = 1 drawn from any point on the line 2x + y = 4 passes through a fixed point.also,find the co-ordinates of that point.


    i could not start the problem
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,807
    Thanks
    116
    Quote Originally Posted by grgrsanjay View Post
    prove that chord of contact of the pair of tangents to the circle x^2 + y^2 = 1 drawn from any point on the line 2x + y = 4 passes through a fixed point.also,find the co-ordinates of that point.


    i could not start the problem
    1. I've made a sketch of the situation (see attachment).
    D and E are the tangent points of the tangents from A to the circle.

    2. For a start have a look here: Pole and polar - Wikipedia, the free encyclopedia
    Attached Thumbnails Attached Thumbnails includes circle,chord and tangent concepts-polareankreis.png  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member grgrsanjay's Avatar
    Joined
    May 2010
    From
    chennai,tamil nadu
    Posts
    143
    Thanks
    1
    i need to prove this theoritically
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,974
    Thanks
    1121
    Did you even look at the websited linked to by earboth? Lots of "theory" there!

    earboth's picture was to help you get started, not give you the answer.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member grgrsanjay's Avatar
    Joined
    May 2010
    From
    chennai,tamil nadu
    Posts
    143
    Thanks
    1
    i could not understand still how to solve problem......
    i could not the theory even sorry

    please help
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by grgrsanjay View Post
    i could not understand still how to solve problem......
    i could not the theory even sorry

    please help
    Maybe you will be allowed to quote certain theorems.
    Your problem can be more easily solved then.

    The .pdf attachment shows one way of seeing the geometry.
    I doubt somehow that you need to go to those lengths,
    but hopefully it is of some help in spite of being general.
    Attached Thumbnails Attached Thumbnails includes circle,chord and tangent concepts-point-intersection-chords.pdf  
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Once you have a suitable proof that all points on the line
    form pairs of tangents to the circle, whose chords have a common point of intersection
    within the circle, you can find that point using similar triangles.
    Attached Thumbnails Attached Thumbnails includes circle,chord and tangent concepts-point-intersection.jpg  
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Alternatively,

    the point Q is closest to the circle.
    The line perpendicular to y=-2x+4, which passes through the circle centre is 2y=x.

    The point of intersection of 2y=x and y=-2x+4 is (1.6, 0.8).

    The equation of the chord of contact from Q is x_qx+y_qy=1 for \left(x_q,\ y_q\right)=(1.6,\ 0.8)=\left(\frac{8}{5},\ \frac{4}{5}\right)

    \frac{8}{5}x+\frac{4}{5}y=1\Rightarrow\ 8x+4y=5

    The point of intersection of 2y=x and 8x+4y=5 gives us the co-ordinates of the point we are looking for.

    Also, if we pick an arbitrary point \left(x_1,\ y_1\right) on the line y=-2x+4,

    then we can write a general equation for all of the chords of contact from points on this line to the circle.

    x_1x+y_1y=1,\;\;\;\;for\;\;y_1=-2x_1+4

    x_1x+\left(-2x_1+4\right)y=1

    \displaystyle\ y=\frac{1-x\left(x_1\right)}{4-2x_1}

    Since all of these chords of contact have different slopes,
    if they all intersect 2y=x at a common point, then the proof is complete.

    y=\frac{1}{2}x

    \displaystyle\frac{1}{2}x=\frac{1-x\left(x_1\right)}{4-2x_1}

    \displaystyle\frac{1}{2}x\left(4-2x_1\right)=1-x_1x\Rightarrow\ 2x-x_1x=1-x_1x

    2x-1=0\Rightarrow\ x=\frac{1}{2}

    Therefore, all the chords have a common point of intersection.
    Attached Thumbnails Attached Thumbnails includes circle,chord and tangent concepts-common-chords.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. circle chord and tangent
    Posted in the Geometry Forum
    Replies: 3
    Last Post: September 23rd 2010, 08:44 AM
  2. Tangent-chord theorem
    Posted in the Geometry Forum
    Replies: 2
    Last Post: April 10th 2010, 07:51 AM
  3. Chord and Circle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: February 7th 2010, 07:22 PM
  4. circle/tangent/chord..help!
    Posted in the Geometry Forum
    Replies: 1
    Last Post: February 3rd 2010, 01:08 AM
  5. Circle and Chord
    Posted in the Geometry Forum
    Replies: 1
    Last Post: March 23rd 2007, 05:10 AM

Search Tags


/mathhelpforum @mathhelpforum