When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct?

Question

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct? I think, if $$rank\left(\begin{array}{cc} a & b \\ a' & b' \end{array} \right)\neq 1$$ then $r_1$ and $r_2$ are distinct. Is correct? Thanks in advance!

Answer 1

case1: rnk=1 and c/c′ in not an integer, case2: rnk≠1 and c=c′. In the first case they are parallel and distint and in the second case they are not parallel and distinct. Consider any two lines they should be parallel ,coincide or not parallel. Thus these two are the only possibibilities for r1 and r2 to be distinct.