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Math Help - Find the equation and comman chord of a circle

  1. #1
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    Find the equation and comman chord of a circle

    Find the equation and length of common chord of the circle
    <br />
2x^2 +  2y^2 + 7x -5y + 2 = 0<br />
and <br />
x^2 + y^2 - 4x + 8y - 18 =0<br />
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  2. #2
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    First, solve for y in each of the equations. Start by moving all x terms and constants to the right side

    2y^2-5y=-2x^2-7x-2

    complete the square
    divide by the coefficient of the quadratic term

    (y-\frac{5}{4})^2=-x^2-\frac{7}{2}x-1+\frac{25}{16}

    taking the root and adding

    y=\pm{\sqrt{-2x^2-\frac{7}{2}x-1+\frac{25}{16}}}+\frac{5}{4}

    Do the same with the other equation and solve the system.
    Last edited by Jameson; May 5th 2009 at 11:37 AM.
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  3. #3
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    Quote Originally Posted by zorro View Post
    Find the equation and length of common chord of the circle
    <br />
2x^2 +  2y^2 + 7x -5y + 2 = 0<br />
and <br />
x^2 + y^2 - 4x + 8y - 18 =0<br />
    Something wrong here. Those two circles don't intersect at all, so how can they have a common chord? (The centres of the circles are at (2,–4) and \bigl(-\tfrac74,\tfrac54\bigr). The distance between these points is about 6.86. This is a lot more than the sum of the two radii, which are \sqrt2 and \sqrt{58}/4, both of them less than 2.)

    Edit. Oops! First radius is \sqrt{38}, not \sqrt2. Thanks to skeeter for spotting that.
    Last edited by Opalg; May 5th 2009 at 11:45 PM. Reason: Bad mistake!
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  4. #4
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    uhh ... the two circles do intersect.
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  5. #5
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    Quote Originally Posted by zorro View Post
    Find the equation and length of common chord of the circle
    <br />
2x^2 +  2y^2 + 7x -5y + 2 = 0<br />
and <br />
x^2 + y^2 - 4x + 8y - 18 =0<br />

    2x^2 +  2y^2 + 7x -5y + 2 = 0

    -2(x^2 + y^2 - 4x + 8y - 18 =0)

    ---------------------------------

    15x - 21y + 38 = 0

    linear equation is the equation of the chord
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    what about the length of the chord ????
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  7. #7
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    Quote Originally Posted by zorro View Post
    what about the length of the chord ????
    You know the equation of the chord and so can find the points where that chord intersects either of the circles (it intersects both circles in the same points, of course). Find the distance between those two points.
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