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Math Help - Circle and Parabola

  1. #1
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    Exclamation Circle and Parabola

    Consider two parabolas
    &

    Find the equation of circle of minimum radius touching both the parabolas.
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  2. #2
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    Quote Originally Posted by sparkle2009 View Post
    Consider two parabolas
    &

    Find the equation of circle of minimum radius touching both the parabolas.
    p_1: y-3=(x+1)^2

    p_2: x-3=(y+1)^2

    p_2 is the image of p_1 of a reflection about the line with the equation y = x. Thus the circle in question touches the parabolae at those points where the gradient is +1.

    I'll take p_1:

    y = x^2+2x+3~\implies~y'=2x+2

    2x+2=1~\implies~x=-\dfrac12

    Therefore the tangentpoint is T\left(-\dfrac12\ ,\ \dfrac{13}4\right)

    A line perpendicular to the angle bisector of the first quadrant (do say first median?) passing through T crosses this angle bisector in the center of the circle in question:

    y-\dfrac{13}4 = -1\left(x+\dfrac12\right)~\implies~y=-x+\dfrac{11}4 intersects y = x at

    A\left(\dfrac{11}8\ ,\ \dfrac{11}8\right)

    The radius is the segment r = |\overline{TA}|
    Attached Thumbnails Attached Thumbnails Circle and Parabola-beruehrkrs_2parabln.png  
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    Quote Originally Posted by earboth View Post
    p_1: y-3=(x+1)^2

    p_2: x-3=(y+1)^2

    p_2 is the image of p_1 of a reflection about the line with the equation y = x. Thus the circle in question touches the parabolae at those points where the gradient is +1.
    earboth could you explain this point ?
    It seems to work but I don't manage to understand it
    Thanks !
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    Quote Originally Posted by running-gag View Post
    earboth could you explain this point ?
    It seems to work but I don't manage to understand it
    Thanks !
    1. The question asks you to find the smallest circle which is tangent to the two parabolae. I've drawn some of those tangent circles (see attachment). You'll get the smallest circle if the tangent points are the endpoints of a diameter, that means the radius is the shortest distance between one parabola and the angle bisector of the first quadrant and the diameter must be perpendicular to the angle bisector.

    2. If a circle and a parabola are tangent to each other then they have a common tangent. This tangent must be perpendicular to the radius of the circle. The radius is perpendicular to the angle bisector and thus the tangent must be parallel to the angle bisector. That means the slope of the tangent is equal to the slope of the parabola (in the tangent point) and it is equal to the slope of th angle bisector.

    3. Since I knew the slope of the angle bisector I used this fact to calculate the coordinates of the tangent point at the parabola.
    Attached Thumbnails Attached Thumbnails Circle and Parabola-beruehrkreise_2parabeln.png  
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