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Math Help - Line and hyperbola intersection

  1. #1
    s3a
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    Line and hyperbola intersection

    "Find the coordinates of the points of intersection of the hyperbola defined by ((x-1)^2)/4 - ((y+2)^2)/9 = 1 and the line 5x + 12y = 0."

    This is really an easy question but for some reason I keep getting it wrong so I must be doing a small mistake somewhere and after several repetitions, I still cannot spot my mistake!

    The answer is: (-1.69, 0.71) and (3.06, -1.27)

    If anyone could please do it for me, I'd greatly appreciated it!
    Thanks in advance!
    Last edited by s3a; May 26th 2009 at 11:23 AM.
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  2. #2
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    Quote Originally Posted by s3a View Post
    "Find the coordinates of the points of intersection of the hyperbola defined by (1) ((x-1)^2)/4 - ((y+2)^2)/9 = 1 and the line (2) 5x + 12y = 0."

    This is really an easy question but for some reason I keep getting it wrong so I must be doing a small mistake somewhere and after several repetitions, I still cannot spot my mistake!

    The answer is: (-1.69, 0.71) and (3.06, -1.27)

    If anyone could please do it for me, I'd greatly appreciated it!
    Thanks in advance!
    1. From (2): y = -\dfrac5{12}x

    2. Plug in this term for y into (1):

    \dfrac{(x-1)^2}4-\dfrac{\left( -\dfrac5{12}x + 2  \right)^2}9=1

    3. Expand the brackets:

    \dfrac{x^2-2x+1}4-\dfrac{ \dfrac{25}{144} x^2- \dfrac{20}{12} x + 4}9=1

    4. Multiply both sides by 9 \cdot 144 = 1296

    324x^2-648x + 324 - 25x^2 +240x-576=1296

    and collect like terms:

    299 x^2  - 408x - 252 = 1296~\implies~299x^2 - 408x - 1548 = 0

    5. Now apply the quadratic formula:

    x = \dfrac{204}{299}\pm\dfrac{54\cdot \sqrt{173}}{299}

    with rounded values you get: x = 3.057722768 ~\vee~ x = -1.693174273

    6. Plug in these values into (2) to calculate the y-values.
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