A minor misunderstanding in coordinate geometry 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.
 A: $ax^2+2hxy+by^2$ 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 ($b,m_1,m_2$).
