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Math Help - GR11 Math: finding endpoints of chord within a circle.

  1. #1
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    GR11 Math: finding endpoints of chord within a circle.

    Given...

    1. Point P is on the line 3x+y=26 and is 10 units from the origin. Determine the coordinates of P.

    I've gotten this far...

    So:
    3x+y=26
    x^2+y^2=\sqrt{10}.

    Then y=26-3x.

    Plugging in y...

    x^2+(26-3x)^2=\sqrt{10}

    10x^2-156x+676=\sqrt{10}

    This is where I'm stuck since I cannot factor it. When the quadratic formula is applied, the number inside the brackets is a negative number.

    2.
    Given...

    From a lighthouse, the range of visibility on a clear day is 40km. On a coordinate system, where O(0,0) represents the lighthouse, a ship is travelling on a course represented by y=2x+80. Between which two points on the course can the ship be seen from the lighthouse?

    I got this far...

    y=2x+80
    x^2+y^2=\sqrt{40}

    Plugging it in...

    x^2+(2x+80)^2=\sqrt{40}

    5x^2+320x+6400=\sqrt{40}

    Again, I am stuck with the same problem. I get a negative number when the quadratic formula is applied.

    What am I doing wrong here? Thank you so much!

    N.
    Last edited by nathan02079; May 6th 2009 at 08:43 PM.
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  2. #2
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    Quote Originally Posted by nathan02079 View Post
    Given...

    1. Point P is on the line 3x-y=26 and is 10 units from the origin. Determine the coordinates of P.

    I've gotten this far...

    So:
    3x+y=26


    Are these signs supposed to be different?
    Last edited by pickslides; May 6th 2009 at 07:45 PM. Reason: typo
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  3. #3
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    Hello, nathan02079!

    In your "distance formula", you're square-rooting when you should square.


    2. From a lighthouse, the range of visibility on a clear day is 40 km.
    On a coordinate system, where O(0,0) represents the lighthouse,
    a ship is travelling on a course represented by: y\:=\:2x+80.
    Between which two points on the course can the ship be seen from the lighthouse?
    Code:
                            |
                            |     *
                            | *
                      Q   * |
                      o     |
                  *    *    |
          P   *         *   |
          o              *  |
      *         *         * |
                      *    *|
      - - - - - - - - - - - * - - - -
                            |O

    P and Q have coordinates (x,y) such that: . OP = OQ = 10.

    We want: . x^2 + y^2 \:=\:10^2

    So we have: . x^2 + (2x+80)^2 \:=\:10^2 \quad\Rightarrow\quad 5x^2 + 320x + 4800 \:=\:0

    Factor: . 5(x+40)(x+24) \:=\:0

    Then: . x \:=\:\text{-}40,\:\text{-}24 \quad\Rightarrow\quad y \:=\:0,\:32


    The ship can be seen from (\text{-}40,\:0) to (\text{-}24,\:32)

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  4. #4
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    Quote Originally Posted by pickslides View Post


    Are these signs supposed to be different?
    Whoops, typo there. :/

    Quote Originally Posted by Soroban View Post
    Hello, nathan02079!

    In your "distance formula", you're square-rooting when you should square.


    Code:
                            |
                            |     *
                            | *
                      Q   * |
                      o     |
                  *    *    |
          P   *         *   |
          o              *  |
      *         *         * |
                      *    *|
      - - - - - - - - - - - * - - - -
                            |O

    P and Q have coordinates (x,y) such that: . OP = OQ = 10.

    We want: . x^2 + y^2 \:=\:10^2

    So we have: . x^2 + (2x+80)^2 \:=\:10^2 \quad\Rightarrow\quad 5x^2 + 320x + 4800 \:=\:0

    Factor: . 5(x+40)(x+24) \:=\:0

    Then: . x \:=\:\text{-}40,\:\text{-}24 \quad\Rightarrow\quad y \:=\:0,\:32


    The ship can be seen from (\text{-}40,\:0) to (\text{-}24,\:32)

    Wow, duh! Thanks for stating the obvious! Darn, I never pay attention.

    Thank you!

    N.

    SOLUTION #1:

    10x^2-156x+576=0
    x=9.6, 6

    Points of intersection: (9.6, -2.8), (6,8)

    SOLUTION #2:

    5x^2+320x+4800=0
    x=-24, -40

    Points of intersection: (-24, 32), (-40,0)

    Thanks again!
    Last edited by nathan02079; May 6th 2009 at 08:52 PM. Reason: Solved
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