Results 1 to 3 of 3

Math Help - geomentry

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    1

    geomentry

    i have two circles with intersecting each other. two circles have same radius.they intersects in two points. i like find formula for finding that two points
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Quote Originally Posted by rattyratty07 View Post
    i have two circles with intersecting each other. two circles have same radius.they intersects in two points. i like find formula for finding that two points
    If you have the circles in a Cartesian form, eg. ax^2+by^2 = r^2 and cx^2+dy^2 = r^2 where r is the radius, you know that at the two intersecting points the coordinates of the two circles are the same.

    Also you know that the radius are the same, therefore you can write it as:

    ax^2+by^2 = cx^2+dy^2 and solve for x,y.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,971
    Thanks
    1637
    Quote Originally Posted by craig View Post
    If you have the circles in a Cartesian form, eg. ax^2+by^2 = r^2 and cx^2+dy^2 = r^2 where r is the radius, you know that at the two intersecting points the coordinates of the two circles are the same.

    Also you know that the radius are the same, therefore you can write it as:

    ax^2+by^2 = cx^2+dy^2 and solve for x,y.
    This is an incorrect formula. " ax^2+ by^2= r^2" is the equation of an ellipse with center at the origin. Since you have two circles, you cannot assume they both have center at the origin so you are better off writing them as (x-a)^2+ (y- b)^2= r^2 and (x- c)^2+ (y- d)^2= r^2 (with same radius). You can solve those two equations for x and y, in terms of a, b, c, and d, to find the points at which they intersect.

    But a more geometric method and, I think, simpler is this: Because the two circles have the same radius, the line between the two points of intersection is the perpendicular bisector of the line between the two centers. Knowing the coordinates of the centers, you can find the coordinates of the midpoint and the slope, m, of the line between centers. The perpendicular bisector is the line through that midpoint with slope -1. Find the points at which that line intersects either circle.

    In fact, here is what I woud do: Given that one circle has center (x0,y0) and radius r and the other has center (x1, y1) and radius r, I would first translate so that the first circle has center at (0,0), by subtracting x0 from every x coordinate and y0 from every y coordinate. That means that the second circle has center at (a, b) where a= x1- x0 and b= y1- y0. The line between centers is now just y= (b/a)x, which has slope b/a, and the midpoint is (a/2, b/2). The perpendicular bisector is given by y= (-a/b)x. Also the first circle now has equation x^2+ y^2= r^2 so the line y= (-b/a)x will intersect the first circle where x^2+ (b^2/a^2)x^2= x^2(1+ b^2/a^2) = \frac{a^2+ b^2}{a^2} x^2= r^2. Now x^2= \frac{a^2r^2}{a^2+ b^2} and x= \pm\frac{ar}{\sqrt{a^2+ b^2}}. Of course, y, then, is y= (b/a)x= \pm\frac{br}{\sqrt{a^2+ b^2}}, using the same sign as the corresponding y: \left(\frac{ar}{\sqrt{a^2+ b^2}}, \frac{br}{\sqrt{b^2+ a^2}}\right) and \left(\frac{-ar}{\sqrt{a^2+b^2}}, \frac{-br}{\sqrt{b^2+ a^2}}\right).

    Finally, translate back by adding x0 to the x coordinate and y0 to the y coordinate.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. geomentry
    Posted in the Geometry Forum
    Replies: 2
    Last Post: September 1st 2009, 12:29 PM
  2. geomentry
    Posted in the Geometry Forum
    Replies: 1
    Last Post: August 20th 2009, 11:39 AM

Search Tags


/mathhelpforum @mathhelpforum