Classification of figure from the general equation of conics

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during my study on conics , I found a formula in the book regarding the classification of figure from the general equation of conics

ax2+2hxy+by2+2gx+2fy+c=0

it was given that
[itex]\Delta=abc+2fgh-af^{2}-{bg}^{2}-{ch}^{2}[/itex]

[itex]if \Delta \neq 0[/itex]
then if
[itex]h^{2}=ab.......parabola[/itex]
[itex]h^{2}<ab......ellipse[/itex]
[itex]h^{2}>ab.......hyperbola[/itex]

if
[itex]\Delta <0..........circle ,h=0,a=b\neq 0,g^{2}+f^{2}-ac>0[/itex]

if
[itex]\Delta = 0[/itex]
if
[tex]h^{2}>=ab..........line[/tex]
[tex]h^{2}<ab..........unique point[/tex]



No explanation regarding the derivation of result was given
neither i could find it on net

hope someone knows it........

thanks in advance
 
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Answers and Replies

  • #2
QuantumQuest
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A full analysis and explanation would require a long document so I'll give an online resource at projecteuclid.org which investigates by elementary algebra the values of ##y## as ##x## increases from ##-\infty## to ##+\infty## and the values of ##x## as ##y## increases from ##-\infty## to ##+\infty## and classifies the loci by their principal graphical properties so found.
 

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