In S.L. Loney's book on The Elements of Coordinate Geometry, the author so educates the reader to find the angle between the two straight lines given by the equation $ax² + 2hxy + by²$. His method is as follows:
Let the seperate equations to the two lines be $y – m_1x$ and $y – m_2x$. So that $ax² + 2hxy + by²$ must be equivalent to $b(y – m_1x)(y – m_2x)$. The question is short and simple (I believe); what is the use of $b$? I don't see how it adds to the proof. I believe what Loney was attempting to make it manifest that multiplying the straight lines by any such constant $b$ wouldn't change their geometrical graph, but other than that I don't see it's purpose. For wouldn't it be like saying that the general equation to the straight line is $b(Ax + By + C) = 0$. If someone could elucidate a much better explanation, I would be ever so greatful. Thank you in advance.
$ax^2+2hxy+by^2$ isn't just "equivalent" to $b(y – m_1x)(y – m_2x)$; the two formulas are exactly equal when $m_1$ and $m_2$ are set to the required values:
$$ ax^2+2hxy+by^2 = b(y – m_1x)(y – m_2x). $$
I suppose the idea is that we start with the equation of the figure written in the form of a general quadratic in $x$ and $y,$ which means we start with $ax^2+2hxy+by^2 = 0$. Then if $b \neq 0,$ you can eliminate $b$ from the expression by dividing by $b,$ and you can do this either before or after factoring. Loney chooses to do it after. If we say, "WLOG let $b=1$," that is essentially doing the division by $b$ before, but it might confuse some less perceptive readers.
Not having a copy of the book handy, I don't know if Loney mentions that this particular factoring of the quadratic is achievable only if $b \neq 0.$ If that is not mentioned, it is a flaw in the argument.
This is not at all the same as writing the equation of a general line as $b(Ax + By + C) = 0.$ In that case you really have introduced an extraneous factor that had no origin in the original formula. If we were to try such a factoring for the equation of a general line, we might instead do this:
$$ Ax + By + C = C(px + qy + 1). $$
This is a correct factoring only if $C \neq 0,$ because we need to set $p = A/C$ and $q = B/C.$ But it starts with three arbitrary parameters ($A,B,C$) and ends with three arbitrary parameters ($C,p,q$). Likewise, Loney's factoring starts with three arbitrary parameters ($a,h,b$) and ends with three ($b,m_1,m_2$).