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Focusing Upon The Maximum (Posted on 2007-12-31) Difficulty: 3 of 5
Determine the maximum area of the circle which is entirely contained within the parabola y2 = 36x, and passes through its focus.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution Solution | Comment 1 of 2

The parabola,
   y^2 = 36x = 4px,
where p is the distance from the vertex to the focus.
The circle,
   (x - h)^2 + y^2 = r^2.
Since the circle passes throught the focus,
   (p - h)^2 + 0^2 = r^2.
   (x - h)^2 + y^2 = (p - h)^2.
The intersection of the circle and ellipse,
   (x - h)^2 + 4px = (p - h)^2
   x^2 - 2(h - 2p)x - (p^2 - 2ph) = 0.
Solving for x,
   x = h - 2p +- sqrt[(h - p)(h - 5p)].
If the radicand is zero, then the circle is
tangent to the ellipse and we have the largest
circle within the ellipse.
h = p implies the radius of the circle is zero.
Therefore, h = 5p and r^2 = (4p)^2.
For our problem,
   Area of circle = PI*(36)^2.

  Posted by Bractals on 2007-12-31 12:26:12
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