Results 1 to 4 of 4

Math Help - Another one on ellipse

  1. #1
    Senior Member
    Joined
    Jul 2006
    From
    Shabu City
    Posts
    381

    Another one on ellipse

    A line segment of length 12 moves with its ends always touching the coordinate axes. Find the equation on the graph of the point on the segment that is 4 units from the end in contact with the x - axis.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    A line segment of length 12 moves with its ends always touching the coordinate axes. Find the equation on the graph of the point on the segment that is 4 units from the end in contact with the x - axis.
    The title "Another one on ellipse" is the clue. Without playing yet, the locus of that particular point is an ellipse.
    After some playing with the line segment and the two coordinate axes, yes, the locus is an ellipse.
    If the line segment is vertical, the point is 4 units from the x-axis.
    Sliding the line segment to the left, when it is horizontal, the point is 8 units from the y-axis.
    Then, sliding the line segment down, when it is vertical again, the point is 4 units from the x-axis.
    Then, sliding the line segment to the right, when it is horizontal again, the point is 8 units from the y-axis.
    Then, sliding the line segment up, when it is vertical, it is in the same position before, the point is 4 units from the x-axis.

    So, the ellipse is "horizontal" or the major axis is along the x-axis.
    major axis = 8+8 = 16 units long.
    minor axis = 4+4 = 8 units high.

    A standard equation of an ellipse centered at the origin (0,0), whose major axis, 2a, is along the x-axis, and whose minor axis, 2b, is then along the y-axis is
    (x^2)/(a^2) +(y^2)/(b^2) = 1

    Therefore the equation of the locus of that particular point is:
    (x^2)/(8^2) +(y^2)/(4^2) = 1
    (x^2)/64 +(y^2)/16 = 1 -----------------answer.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,905
    Thanks
    765
    Hello, ^_^Engineer_Adam^_^!

    It took me several tries, but I found a painless approach . . .


    A line segment of length 12 moves with its ends always touching the coordinate axes.
    Find the equation on the graph of the point on the segment that is 4 units
    from the end in contact with the x - axis.
    Code:
                |
           (0,b)*
                |   *
                |       *
                |           *  P(x,y)
                + - - - - - - - o
                |               :   *
          ------+---------------+-------*--
                |                     (a,0)

    Let the intercepts of the segment be (a,0) and (0,b).
    . . Then: . a^2 + b^2\:=\:12^2 [1]

    From similar triangles we have: . \begin{Bmatrix}x = \frac{2}{3}a\quad\Rightarrow\quad a = \frac{3}{2}x \\ \\<br />
y = \frac{1}{3}b\quad\Rightarrow\quad b = 3y \end{Bmatrix}

    Substitute into [1]: . \left(\frac{3}{2}x\right)^2 + (3y)^2\:=\:144\quad\Rightarrow\quad \frac{9x^2}{4} + 9y^2\;=\;144

    Divide by 144\!:\;\;\boxed{\frac{x^2}{64} \,+ \,\frac{y^2}{16}\;=\;1} . . . ta-DAA!

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    A line segment of length 12 moves with its ends always touching the coordinate axes. Find the equation on the graph of the point on the segment that is 4 units from the end in contact with the x - axis.
    Hi,

    I've costructed your problem by hand (of course I'm kidding ) and recorded the changes. In the attachment you can see the result of hours of constructing (yes, I'm kidding again)

    EB
    Attached Thumbnails Attached Thumbnails Another one on ellipse-ellips_prod2.gif  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ellipse
    Posted in the Geometry Forum
    Replies: 1
    Last Post: November 17th 2010, 07:48 AM
  2. ellipse
    Posted in the Geometry Forum
    Replies: 1
    Last Post: November 16th 2010, 12:29 PM
  3. ellipse 2
    Posted in the Geometry Forum
    Replies: 1
    Last Post: November 15th 2010, 12:41 PM
  4. ellipse
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: June 7th 2010, 05:08 PM
  5. Ellipse
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: March 17th 2008, 06:54 AM

Search Tags


/mathhelpforum @mathhelpforum