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November 23rd, 2009, 04:50 PM   #1
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Completing the square

Hello,

Can someone please help me with this problem?
I need to determine whether it is an ellipse, hyperbola, parabola, or degenerate conic.

x^2 - y^2 =10 (x-y) + 1

Thanks!
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November 23rd, 2009, 05:30 PM   #2
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Hint: it can be written as (x - 5) - (y - 5) = 1.
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November 23rd, 2009, 05:42 PM   #3
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Re:

Quote:
Originally Posted by skipjack
Hint: it can be written as (x - 5) - (y - 5) = 1.
Well, that's true... but I'm guessing "flip" doesn't know that.

Here's a little flow-chart.
1) Is either x^2 or y^2 missing? If so, it is a parabola. Otherwise...
2) Are the signs of the x^2 and y^2 terms different (when written on the same side of the equation)? If so, it's a hyperbola. Otherwise...
3) Are the coefficients of x^2 and y^2 the same ("exactly" the same, in case that was obvious)? If so, it's a circle. Otherwise...
4) It's an ellipse.

There are complicated tests for degenerate conics that involve determinants and so forth. As a good rule of thumb, I would recommend writing the equation in "standard" form (like skip did) and looking for conspicuous zeros, or negative numbers on the right hand side of a circle/ellipse equation.
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November 23rd, 2009, 06:24 PM   #4
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Re: Completing the square

Thanks for the help!
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