Posted by sue on Thursday, December 30, 2010 at 4:12am.
In all of these problems complete the square, then look at standard form of conic section.
For example to find the center of the ellipse
Find the center of an ellipse 9x^2+4y^2-36x-24y-36=0
9 x^2 -36 x -36 = -4 y^2 + 24 y
(-9/4) x^2 + 9 x + 9 = y^2 -6 y
(-9/4) x^2 + 9 x + 18 = y^2 -6 y + 9
(-9/4) x^2 + 9 x + 18 = (y-3)^2
x^2 - 4 x - 8 = (-4/9) (y-3)^2
x^2 - 4 x = (-4/9) (y-3)^2 + 8
x^2 - 4 x + 2 = (-4/9) (y-3)^2 + 10
(x-2)^2 + (4/9) (y-3)^2 = 10
You can see right now that center is at (2,3)
(x-2)^2/10 + (y-3)^2/(90/4) = 1
x^2 - 4 x = (-4/9) (y-3)^2 + 8
x^2 - 4 x + 4 = (-4/9) (y-3)^2 + 12
(x-2)^2 + (4/9) (y-3)^2 = 12
You can see right now that center is at (2,3)
(x-2)^2/12 + (y-3)^2/(27) = 1
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