Identify the Conic and put it into Standard Form
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Identify the Conic and put it into Standard Form

[From: ] [author: ] [Date: 12-04-10] [Hit: ]
and I seem to catch on to this pretty easily. If you could post an explanation of how you got your answer that would be great, it does not have to be long.Also, if you have some dependable resources on where I can learn more about Standard Form Conic Sections, that would be very helpful to me.......
Find the standard form of the following conic section --- 9x^2 + 4y^2 - 18x - 24y + 9 = 0
As far as I know, this is either a Hyperbola or an ellipse, perhaps a parabola. I know their standard forms but am unsure how to get there or how to identify the conic.

Now this shouldn't be too hard, and I seem to catch on to this pretty easily. If you could post an explanation of how you got your answer that would be great, it does not have to be long.
Also, if you have some dependable resources on where I can learn more about Standard Form Conic Sections, that would be very helpful to me.

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This should be an ellipse, as the squared terms are added, and have different coefficients.
A circle has the same coefficient on the x^2 and y^2.
An hyperbola would have a minus sign between the squared terms.
A parabola only has one squared term.

9x^2 + 4y^2 - 18x - 24y + 9 = 0

Regroup the terms:
9x^2-18x +4y^2-24y = -9

Factor out the lead coefficients on the squared terms:
9(x^2-2x) +4(y^2-6y)= -9

Complete the squares: be sure to multiply the added terms by the lead coefficients.
9(x^2-2x +1) + 4(y^2-6y +9) = -9+ 9(1)+ 4(9)

Factor:
9(x-1)^2+ 4(y-3)^2 = 36

Divide by 36
(x-1)^2 + (y-3)^2 = 1
---------- ----------
4 ...... .....9

Hoping this helps!

Most analytic geometry and precalculus textbooks have a good explanation of conics.
Good luck!

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9x² + 4y² - 18x - 24y + 9 = 0

9x² -18x + 4y² - 24y = - 9

(9x² - 18x) + (4y² - 24y) = - 9

9(x² - 2x) + 4(y² - 6y) = - 9

9(x² - 2x + 1) + 4(y² - 6y + 9) = - 9 + 9(1) + 4(9)

9(x - 1)² + 4(y - 3)² = - 9 + 9 + 36

9(x - 1)² + 4(y - 3)² = 36

[9(x - 1)² / 36] + [4(y - 3)² / 36] = 36 / 36

[(x - 1)² / 4] + [(y - 3)² / 9] = 1



Ellipse

Center (1, 3)

Major Axis: y = 3

Minor Axis: x = 1

Vertices (- 2, 3), (4, 3), (1, 5) & (1, 1)

Focii (- 1.24, 3) & (3.24, 3)

Length = 6

Width = 4
 
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