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

ax

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

ax

^{2}+2hxy+by^{2}+2gx+2fy+c=0it 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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