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Question 281466: state which type of conic section each equation represents. then put the equation into standard form.
x^2+9y^2-4x+54y+49=0
Answer by Mathematicians(84)   (Show Source):
You can put this solution on YOUR website! 
The key points on this on is that you have a positive  and a positive  , you only have two choices on what this could be, either a circle or ellipse. However, the coefficient near the y^2 has a 9 while the x^2 has a 1, so it cannot be a circle (otherwise they would be the same).
To put it in standard form, you will heavily use the completing the square method.
 subtract 49 on both sides
 group the x's and y's together
You will notice I put +_ and -_ in each of those answers, that is because whenever you add and subtract the same number, you get 0, so you can do that in this case. The number you will add will be whatever half the term near the x or y and square it. For example, for the 4x  I got the 9 doing the same method as above. From here, we can add 4 to both sides and add 81 to both sides. Why 81? Well the -9 is inside parenthesis and in order to get it out, you need to distribute the 9 from outside then add it over to the right side
Finally do some factoring:
Divide 36 on both sides and you get:
Which simplifies to
This is in standard form, you didn't need to write the 6^2, this makes it easier to see the major and minor axis of the ellipse.
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