ILL 1985 Problem 47

Let $F$ be the correspondence associating with every point $P = (x, y)$ the point $P' = (x', y')$ such that $$x'= ax + b, y'= ay + 2b. (1)$$ Show that if $a \neq 1$ , all lines $PP'$ are concurrent. Find the equation of the set of points corresponding to $P = (1, 1)$ for $b = a^2$ . Show that the composition of two mappings of type $(1)$ is of the same type.