Q.Find the joint equation of the pair of lines through the point $(2, -3)$ and parallel to lines represented by $x^2 + xy - y^2 = 0$.

If $u \equiv a_1x+b_1y+c_1$ and $v \equiv a_2x+b_2y+c_2$ are two lines, their product $uv=0$ is called the combined (or joint) equation of the pair: it is a single equation whose solution set is exactly the union of the two lines, because a product of real numbers is zero precisely when at least one factor is zero. Building a combined equation from two known lines is just multiplication and expansion, as in $(x+y-2)(2x-y+2)=0$ expanding to $2x^2+xy-y^2-2x+4y-4=0$. Going the other way -- splitting a combined equation back into its separate lines -- means factoring the second-degree expression into two linear factors, e.g. $x^2-y^2+x+y=(x+y)(x-y+1)=0$ gives the lines $x+y=0$ and $x-y+1=0$. This idea is the foundation for every other technique in the chapter: a homogeneous pair through the origin, a general second-degree pair, and lines constructed to be parallel or perpendicular to a given pair are all combined equations of this same u·v = 0 form.