From the joint equation of a pair of straight lines $ax^2+2hxy+by^2+2gx+2fy+c=0$, is it possible to determine whether the two lines are intersecting or not, without breaking (or factorising) the joint equation to separate linear equations?
I know to find the angle between the lines by the following formula $$\tan \theta=\left|\frac{2\sqrt{h^2-ab}}{a+b} \right|$$ without finding the individual equations of the lines. Similarly, is there any algorithm to determine whether the lines intersect or not from the joint equation directly?