In Jürgen Richter-Gebert's book "Perspectives on Projective Geometry", he talks about Plucker’s $\mu$ in Section 6.3. He says that this trick was used by Plucker quite often.
Plucker's trick involves finding the equation of a curve in a plane which passes through the intersection of two geometric objects of the same type. For example, if $f(x,y) = ax+by+c$, then $f \equiv 0$ represents a straight line. If we want the equation of a line passing through the intersection of two lines $f_1$ and $f_2$ and a point $(u,v)$, then according to Plucker, it is evidently $$f_1(u,v)f_2(x,y) - f_1(x,y)f_2(u,v).$$
One can do this for conics as well. If $f_1$ and $f_2$ represents two conic equations, then the above form represents the conic passing through the intersections of the two conics and passing through the point $(u,v)$.
I want to know some more applications of this trick. For example, if two conics do not intersect at four points, Plucker's trick yields a conic. What is the meaning of this conic?
As a special case, if two circles do not intersect, then Plucker's trick gives a circle passing through the point $(u,v)$. But I am not able to understand the significance of such a circle.