Combined Equation of a Pair of Lines Through the Origin

Two straight lines through the origin can always be written as $l_1x + m_1y = 0$ and $l_2x + m_2y = 0$, since any line through the origin has an equation with zero constant term. Multiplying the two equations together gives a single equation satisfied by every point that lies on either line:

Expanding the left side,

Writing $a = l_1l_2$, $2h = l_1m_2 + l_2m_1$, $b = m_1m_2$, this becomes

which we call the combined equation of the pair of lines. Notice that every term is of degree exactly 2 in $x$ and $y$ — such an equation is called a homogeneous equation of second degree. This is the algebraic signature of "a pair of lines through the origin": a point $(x,y)$ satisfies $ax^2+2hxy+by^2=0$ exactly when it lies on the first line, the second line, or both, because the product of two real numbers is zero only when at least one factor is zero.

A useful way to see this concretely: take the lines $x - 2y = 0$ and $2x + y = 0$. Their product is

so $a=2$, $2h=-3$ (i.e. $h=-\tfrac32$), $b=-2$ is the combined equation of this particular pair. Given the combined equation, we recover the individual lines by factorising the quadratic expression back into two linear factors — this is the reverse of the construction above, and it is always possible over the reals only under a condition on $a,h,b$ that we examine in the next section.

It is worth stressing what this equation is not: it is not the equation of a curve passing "between" the two lines, nor a single new line. It is one polynomial equation whose solution set is exactly the union of the two lines, which is why we call it "combined." This idea — that the product of two linear expressions gives one quadratic expression whose zero set is the union of the zero sets of the factors — is the entire foundation on which every other result in this chapter (the angle between the lines, their bisectors, the general non-origin case, and the homogenising technique) is built, so it deserves to be genuinely understood before moving on, rather than only memorised as a formula to substitute into.