Results 1 to 2 of 2

Math Help - ellipse problem

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

    ellipse problem

    A point moves in the coordinate plane so that the sum of its distances from (1,2) and (5,2) is equal to 16. Find the equation of the path of the moving point...

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,903
    Thanks
    329
    Awards
    1
    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    A point moves in the coordinate plane so that the sum of its distances from (1,2) and (5,2) is equal to 16. Find the equation of the path of the moving point...

    This is the classic problem to derive the equation of an ellipse with foci at (1, 2) and (5, 2).

    Call a point on the curve (x, y). Then the distance formula says:
    \sqrt{(x - 1)^2 + (y - 2)^2} + \sqrt{(x - 5)^2 + (y - 2)^2} = 16

    Get rid of the radicals:
    \sqrt{x^2 + y^2 - 2x - 4y + 5} + \sqrt{x^2 + y^2 - 10x - 4y + 29} = 16

    \sqrt{x^2 + y^2 - 2x - 4y + 5} = 16 - \sqrt{x^2 + y^2 - 10x - 4y + 29}

    x^2 + y^2 - 2x - 4y + 5 = (16 - \sqrt{x^2 + y^2 - 10x - 4y + 29})^2

    x^2 + y^2 - 2x - 4y + 5 = 256 - 32 \sqrt{x^2 + y^2 - 10x - 4y + 29} + (x^2 + y^2 - 10x - 4y + 29)

    32 \sqrt{x^2 + y^2 - 10x - 4y + 29} = 280 - 8x = 8(35 - x)

    4 \sqrt{x^2 + y^2 - 10x - 4y + 29} = 35 - x

    16 (x^2 + y^2 - 10x - 4y + 29) = (35 - x)^2

    16x^2 + 16y^2 - 160x - 64y + 464 = 1225 - 70x + x^2

    15x^2 - 90x + 16y^2 - 64y = 761

    Now complete the square on x and y:
    15(x^2 - 6x) + 16(y^2 - 4y) = 761

    15(x^2 - 6x + 9) + 16(y^2 - 4y + 4) = 761 + 15 \cdot 9 + 16 \cdot 4

    15(x - 3)^2 + 16(y - 2)^2 = 960

    Divide both sides by 960:
    \frac{(x - 3)^2}{64} + \frac{(y - 2)^2}{60} = 1

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ellipse problem
    Posted in the Geometry Forum
    Replies: 3
    Last Post: March 6th 2011, 10:51 PM
  2. ellipse problem
    Posted in the Geometry Forum
    Replies: 0
    Last Post: November 22nd 2010, 08:33 AM
  3. Ellipse problem
    Posted in the Geometry Forum
    Replies: 1
    Last Post: March 30th 2010, 08:09 PM
  4. Ellipse problem
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 7th 2010, 08:16 PM
  5. ellipse problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 16th 2009, 10:04 AM

Search Tags


/mathhelpforum @mathhelpforum