If $ax^2 + 2bxy + by^2 = 0$ represents a pair of lines, then find the combined equation of lines that can be obtained by reflecting these lines about the x-axis.
I know that this can be done by separating the lines from the given equation and then finding the reflected rays and then combining the reflected rays again, but I was wondering if there is a short way of solving this problem?
The reflection about the x-axis can be rapidly obtained substituting $y$ with $-y$. So you obtain $ax^2-2bxy+by^2$. Also note that, in this case, the new lines are symmetrical to the initial Iines with respect to the y-axis as well.