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 Focusing Upon The Maximum (Posted on 2007-12-31)
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 | 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.`
`Therefore,`
`   (x - h)^2 + y^2 = (p - h)^2.`
`The intersection of the circle and ellipse,`
`   (x - h)^2 + 4px = (p - h)^2`
`              or`
`   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 istangent to the ellipse and we have the largestcircle 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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