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November 23rd, 2009, 04:50 PM  #1 
Newbie Joined: Sep 2009 Posts: 9 Thanks: 0  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 (xy) + 1 Thanks! 
November 23rd, 2009, 05:30 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
Hint: it can be written as (x  5)²  (y  5)² = 1.

November 23rd, 2009, 05:42 PM  #3  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Quote:
Here's a little flowchart. 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.  
November 23rd, 2009, 06:24 PM  #4 
Newbie Joined: Sep 2009 Posts: 9 Thanks: 0  Re: Completing the square
Thanks for the help!


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