General Second Degree Equation in x and y

So far every pair of lines considered has passed through the origin (via a homogeneous equation) or been built directly from two given linear factors. This section studies the fully general picture: an equation of the form $ax^2+2hxy+by^2+2gx+2fy+c=0$ (with at least one of $a,b,h$ nonzero) is called a general second-degree equation in $x$ and $y$.

Theorem 4.5. The combined equation of any two lines is a general second-degree equation in $x$ and $y$.

Proof. Let $u \equiv a_1x+b_1y+c_1$ and $v\equiv a_2x+b_2y+c_2$ be two lines, so their combined equation is $uv=0$, i.e. $(a_1x+b_1y+c_1)(a_2x+b_2y+c_2)=0$. Expanding, $a_1a_2x^2 + a_1b_2xy + a_1c_2x + b_1a_2xy + b_1b_2y^2 + b_1c_2y + c_1a_2x + c_1b_2y + c_1c_2 = 0$. Writing $a=a_1a_2$, $b=b_1b_2$, $2h=a_1b_2+a_2b_1$, $2g=a_1c_2+a_2c_1$, $2f=b_1c_2+b_2c_1$, $c=c_1c_2$, this is exactly $ax^2+2hxy+by^2+2gx+2fy+c=0$, the general second-degree form. $\blacksquare$ …