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Math Help - Tangent to a parabola

  1. #1
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    Tangent to a parabola

    Got stuck on another hw problem, can someone help please?

    Show that the line with the equation 4y-3x=20 is a tangent to both the circle with equation x^2+y^2=16 and the parabola with equation y^2=15x.

    I don't know which points the tangent touch the circle and the parabola at, and I'm not sure how to find it.
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  2. #2
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    Someone pointed in out in another thread of ours that it is only necessary to show that there is a single point of intersection to prove tangency. In this case, I think the easiest approach is to take 4y-3x=20 and isolate one of the variables. Next, substitute your result into the circle and find the single point of tangency.

    Repeat the same process with 4y-3x=20 and the parabola.
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  3. #3
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    The line equation is y=\frac{3}{4}x+5

    We have the circle y=\pm\sqrt{16-x^{2}} and the parabola

    y=\pm\sqrt{15x}

    Set the line equation equal to the circle and the parabola, respectively:

    \frac{3}{4}x+5=\sqrt{16-x^{2}}

    Solving for x, we see x=\frac{-12}{5}

    \frac{3}{4}x+5=\sqrt{15x}

    x=\frac{20}{3}

    Now, find the derivative(slope) of the circle and parabola at those points.

    If they both equal 3/4(the slope of the line), then we're in business.
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  4. #4
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    I've done that question now, thanks for the help.

    But I've come across another similar question that doesn't give the equation of the tangent or the point of intersection, so I'm not sure what to do again:

    Find an equation of the line which is tangent to both the parabola y^2=4ax and the parabola x^2=4ay.

    The answer is x+y+a=0, but I've got no idea where that came from. Can I assume that the tangent touches the two parabola's at the same point?

    Help would be greatly appreciated.
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  5. #5
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    Quote Originally Posted by mathdeity View Post
    Someone pointed in out in another thread of ours that it is only necessary to show that there is a single point of intersection to prove tangency.
    Warning: That tangency property only works for circles. It will not work for functions in general. (Though I think it might for a parabola, too. I'll have to think about it. It is best not to assume this.)

    -Dan
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  6. #6
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    This post is wrong, sorry.
    Warning: That tangency property only works for circles. It will not work for functions in general. (Though I think it might for a parabola, too. I'll have to think about it. It is best not to assume this.)
    This property does apply to parabolas, and any other function without a point of inflection.

    The part about it working for parabolas is right though
    Last edited by badgerigar; December 17th 2007 at 04:38 AM. Reason: I was wrong.
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  7. #7
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    Find an equation of the line which is tangent to both the parabola y^2=4ax and the parabola x^2=4ay.

    The answer is x+y+a=0, but I've got no idea where that came from. Can I assume that the tangent touches the two parabola's at the same point?

    Help would be greatly appreciated.
    The tangent does not touch the parabolas (parabolae? neither of these look right. oh well) at the same point.
    What you can do is use the fact that the tangent will have the form y=mx+c. Then you can substitute this for y in both equations and find out what values m and c take if the equations have exactly one solution.
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