Given a line in a 2D space, defined by equation $Ax+By+C=0$, is it possible to find a point of it without assume that A or B are different of 0 ?
That is, usually the method to find some point is "assume $A \ne 0$, if we fix $y=0$ the solution of the equation gives that point $(-C/A,0)$ is a point of the line, otherwise $B \ne 0$ and ...·
But it is possible any other method without split the problem in two ?
In other words, is it possible to find an expression for some point (any one) of the line that doesn't contains a division by A, B or C or by any other term that can be zero in some cases ?
The question could be expressed in another way: given a line $Ax+By+C=0$, give an expression of the same line in vector/parametric form that is valid for any value of A, B, C.
The direction vector is easy to find, (-B,A), the remainder target is to find the expression of some point.