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Math Help - Parabola Question

  1. #1
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    Post Parabola Question

    Find the equation of the locus of the center of the circle which touches the X axis, and also touches the Circle with Center(0,3) and Radius 2 units. Also prove that this locus is a Parabola.
    Please help on how to start the problem.
    My approach was finding 2 points which are definitily present on the locus (by rough plot) and then proving y^2 is proportional to x, but I guess that won't work.
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  2. #2
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    Quote Originally Posted by Programmer View Post
    Find the equation of the locus of the center of the circle which touches the X axis, and also touches the Circle with Center(0,3) and Radius 2 units. Also prove that this locus is a Parabola.
    Please help on how to start the problem.
    My approach was finding 2 points which are definitily present on the locus (by rough plot) and then proving y^2 is proportional to x, but I guess that won't work.
    Let M(m, t) denote the center of the circle which touches the circle around C(0,3;r=2).

    1. Then the distance

    |\overline{CM}|=\sqrt{m^2+(t-3)^2}

    2. According to the question

    |\overline{CM}| - 2 = t

    3. Solve this equation for t:

    t=\dfrac1{10}(m^2+5)

    4. The curve which is produced by all points M is consequently:

    y = \dfrac1{10}(x^2+5)
    Attached Thumbnails Attached Thumbnails Parabola Question-locus_mittelpktberuehrkrs.png  
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  3. #3
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    Quote Originally Posted by Programmer View Post
    Find the equation of the locus of the center of the circle which touches the X axis, and also touches the Circle with Center(0,3) and Radius 2 units. Also prove that this locus is a Parabola.
    Please help on how to start the problem.
    My approach was finding 2 points which are definitily present on the locus (by rough plot) and then proving y^2 is proportional to x, but I guess that won't work.
    Second approach:

    Obviously the parabola in question is:
    - opening upward
    - symmetric to the y-axis

    Therefore the genral equation of such a parabola is:

    y = ax^2+b

    There are 2 points which lay on the parabola: P\left(0\ ,\ \frac12\right) and Q(5,3)

    Plug in the coordinates of these points and solve the system of equations for a and b:

    \left|\begin{array}{rcl}\dfrac12&=&b\\3&=&25a+b\en  d{array}\right.

    You'll get: a = \dfrac1{10}~\wedge~b=\dfrac12
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  4. #4
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    Thanks a lot!
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