conic sections and families

[drufae]

When does the generic equation of the 2nd degree not represent a conic section or does it always represent a conic section (unless it has no possible solutions in real numbers)?

 

The other topic I would like like to devote this thread to is families (or sometime called pencils) of conics.

[math] C_1 + \lambda C_2 = 0 [/math] represents the zero sets of the family of conics passing through the intersection points of conics [math] c_1 and c_2 [/math] please note that [math]\lambda[/math] is a parameter

and assumes all possible real values.

 

[math] C_1 + C_2 = 0 [/math] is the same as the logical operation of this must satisfy both [math] C_1 AND C_2 = 0[/math].however there is an incongruity: suppose [math] C_1|_{(x_1,y_1)} = -1 [/math] and [math] C_2|_{(x_1,y_1)} = 1 [/math]. this still satisfies the original equation of the 2 conics.But [math]{(x_1,y_1)} [/math] does not belong to the zero set of either original conic .i.e. it does not satisfy either of the original conic equations.

 

Furthermore the generated shape will have all the points of both conics plus some extra.thus while the resulting equation is a general equation of the second degree it forms no recognizable conic.

 

Is my reasoning correct or have have really missed out a lot of logic?