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Math Help - Tangent of ellipse problem

  1. #1
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    Tangent of ellipse problem

    it is given that the line y=mx+c is tangent to the ellipse
    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 if a^2m^2=c^2-b^2.
    Show that if the line y=mx+c passes through the point (\frac{5}{4}, 5) and is tangent to the ellipse 8x^2+3y^2=35, then c is \frac{35}{3} or \frac{35}{9}.

    At the point, 5=\frac{5}{4}m+c
    substitute m,
    \frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}
    i rearrange it and get
    776c^2-8960c+22925=0
    but the roots of c are not what are given.
    Thanks
    Last edited by arze; May 8th 2010 at 02:28 AM.
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  2. #2
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    a^2 = \frac{35}{8} ; and ; b^2 = \frac{35}{3}
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  3. #3
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    Quote Originally Posted by sa-ri-ga-ma View Post
    a^2 = \frac{35}{8} ; and ; b^2 = \frac{35}{3}
    yes. isn't that what i am using?
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  4. #4
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    Quote Originally Posted by arze View Post
    yes. isn't that what i am using?
    No. You have interchanged a and b.
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  5. #5
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    Quote Originally Posted by arze View Post
    it is given that the line y=mx+c is tangent to the ellipse
    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 if a^2m^2=c^2-b^2.
    Show that if the line y=mx+c passes through the point (\frac{5}{4}, 5) and is tangent to the ellipse 8x^2+3y^2=35, then c is \frac{35}{3} or \frac{35}{9}.

    At the point, 5=\frac{5}{4}m+c
    substitute m,
    \frac{35}{3}(4-\frac{4}{5}c)^2=c^2-\frac{35}{8}
    i rearrange it and get
    776c^2-8960c+22925=0
    but the roots of c are not what are given.
    Thanks
    hi

    like sarigama suggested , you misplaced both a^2 and b^2

    should be

     <br />
\frac{35}{8}(\frac{20-4c}{5})^2=c^2-\frac{35}{3}<br />
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  6. #6
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    ok, i see what you mean, but shouldn't a be bigger than b in the case of an ellipse?
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  7. #7
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    Quote Originally Posted by arze View Post
    ok, i see what you mean, but shouldn't a be bigger than b in the case of an ellipse?
    Not necessarily.
    If the foci lie on the y axis, a<b.
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