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This appears to be an ellipse. We won't know for sure until we have transformed the equation into the form:
We start by gathering the x terms and y terms together:
Then we factor out coefficients of the squared terms from each set:
Next we figure out what number we would need in each set of parentheses to make that expression a perfect square. This is done by taking 1/2 of the coefficient the x (or y) term and squaring it. The coefficient of the x term is -6. Half of this is -3. -3 squared is 9. So we need a 9 in the first set of parentheses.
For the second set of parentheses, the coefficient of the y term is -8. Half of this is -4 and -4 squared is 16. So we need a 16 in the second set of parentheses.
The tricky part is in knowing how to add the 9 and the 16 into these parentheses correctly. Since the first set of parentheses has a 16 in front of it then when we add a 9 inside, we are actually adding 16*9 to the left side. And if we add 16*9 of the left side, then we must add 16*9 on the right side, too.
With similar logic we can figure out that adding a 16 in the second set of parentheses is really adding a 25*16 to the left side (because of the 25 in front of the parentheses. So here is what we get:
Take a long look at this to see how all this is working. It is the most difficult part of these kinds of problems.
Simplifying the right side we get:
Now we divide both sides by 400 (or multiply by 1/400) to get the 1 we want on the right:
And we can write the numerators as perfect squares:
Since we were able to transform the equation into the proper form, this is indeed an ellipse. And we can "read" the center from the numerators: (3, 4)
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