equation representing 2 straight lines

Question

Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factorize it? Is there a way which we can factorize all such equations which represents two straight lines or it require some special technique ?

Answer 1

I have changed my mind. First factorize $12x^2+7xy-10y^2$ as the product of two factors $(ax+by)(cx+dy)$. It is just like factoring an ordinary quadratic.

Then bring in $e$ and $f$ like this: $(3x-2y+e)(4x+5y+f)$ The linear terms are $e(4x+5y)+f(3x-2y)=13x+45y$ Collect the coefficients of $x$, so $4e+3f=13$, and also an equation for $y$.

You now have two equations in $e$ and $f$ to be satisfied at the same time. If you can solve them, you are done.

As a check, $ef$ should equal -3. If they don't, it was a hyperbola, not a pair of lines.

Answer 2

The general version of the quadratic equation of which you give a special example will represent a conic or a degenerate conic section. There is a long history of using "invariants" associated with the coefficients of this general equation to determine what one has. One discussion of this can be found here: http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node28.html but you can also look up information about "discriminants." Scroll down for some treatment of this on this page: http://en.wikipedia.org/wiki/Conic_section