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 Two Circles And Maximum Triangle (Posted on 2008-01-11)
The circle x2 + y2 = 4 intersects the x axis respectively at the points E and F. A different circle with variable radius with its center located at F cuts the former circle at point G located above x axis and intersects the line segment EF at the point H.

Determine the maximum area of the triangle FHG.

 See The Solution Submitted by K Sengupta No Rating

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 Solution | Comment 1 of 6
`First circle,`
`   x^2 + y^2 = 2^2`
`Second circle,`
`   (x - 2)^2 + y^2 = r^2, where r < 4.`
`Solving these for point G(x,y),`
`        8 - r^2   x = ---------            4`
`        r*sqrt(16 - r^2)   y = ------------------               4`
`Area of triangle FGH,`
`   A = (1/2)(base)(altitude) = (1/2)(r)(y)`
`        r^2*sqrt(16 - r^2)      = --------------------                8`
`Setting dA/dr = 0 and solving for r^2 gives,`
`   r^2 = 32/3`
`Plugging this into the area formula gives.`
`   A = 16*sqrt(3)/9 ~= 3.0792`
`This agrees with Geometer's Sketchpad.`
` `

 Posted by Bractals on 2008-01-11 12:33:02

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