# Confusion with the various forms of the equation of second degree

I am confused with the second degree equation,an equation of second degree $ax^2+by^2+2hxy+2gx+2fy+c=0$ represents a conic,and nature of the conic depends on the various other conditions,like if $\Delta = abc + 2fgh - bg^2 - ch^2 - af^2 = 0$ and $h^2 \ge ab$ then this represent a pair of straight lines,$h^2 \gt ab$ then they are intersecting straight lines, $h^2 = ab$ then parallel straight line .. like this other conditions for circle and ellipses are also given.

But I couldn't not understand how this is happening and no proof is given in my module and hence am not getting the feel of it,for circle and ellipse I could somehow see that if the substitute the equations appropriately I get the required equations of circle and ellipse or hyperbola but things are pretty much confusing for in case of pair of straight lines,my module stretches out the discussion on this by giving some formulas for the angle between the pair of straight lines,equation of the angle bisector then homogeneous form ($ax^2+2hxy+by^2$) area of the triangle formed by $ax^2+2hxy+by^2$ and $lx+my+n=0$ but all this things seems just like chug and plug formulas for me .. I don't want to memorize them just like that as I would forget them easily.

So could anybody suggest a proper reference in this regard?

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There should be a term $by^2$ too, right? – Hans Lundmark Apr 21 '11 at 15:53
@Hans Lundmark:Yes,fixed now. – Quixotic Apr 21 '11 at 16:00