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# Algebra: Ellipses and Circles

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1. Find the center of the ellipse given by 64x2 + 4y2 + 128x - 80y +208 = 0
2. Find the radius of the circle defined by x22 + y2 - 6x -16y = 48
3. The distance between the foci of the ellipse defined by 9(x + 6)2 + 25(y - 15)2 = 225
4. What is the coefficient of x3 in the expansion of (2+ x)5

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Ellipses and circles are analyzed to find radii and distance between foci.

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1. Find the center of the ellipse given by 64x2 + 4y2 + 128x - 80y +208 = 0
We can rewrite the function into:
64(x2+2x+1)-64+ 4(y2- 20y+100)-400+208= 0
64(x+1)2+ 4(y-10) 2=256
(x-(-1))2/4+ (y-10) 2/64=1
then the center of the ellipse is (-1, 10)
and for the ellipse, a=8 and b=2

2. Find the radius of the circle defined by x2 + y2 - 6x -16y = 48
We can rewrite the function ...

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