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well, I have forgotten how to identify ellipse, hyperbola,circle straightline from the general equation of conic, so is there any other way to identify these homeomorphic or not? a) B is an ellipse, b) B is an hyperbola, c) B is an complement of a closed ellipse. please help.

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Those are not in general form (no mixed terms). So you only need to complete squares. – azarel Jul 19 '12 at 20:13
You could plot them on Wolfram Alpha. – Rudy the Reindeer Jul 19 '12 at 20:18
Thank you for the comment of azrael, I have calculated that $(a)$ A is an hyperbola so is not homeomorphic to B, $(b)$ both are hyperbola so homeomorphic, but still confused about $c$ – Un Chien Andalou Jul 19 '12 at 20:19
Hint for (c): $x^2 - 2 x + 1 = (x-1)^2$ – Robert Israel Jul 19 '12 at 20:24
@Patience, please get some graph paper and draw some pictures. This is upsetting. – Will Jagy Jul 19 '12 at 20:38
up vote 4 down vote accepted

A is a circumference in a. and b. while a disk in c. so...

a. Yes, homeomorphism by isometric transformation to overlay the centers, and then by projection of one onto the other.

b. No, the circumference is compact, while the hyperbola is not

c. No, same reason as b

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