Classify a pair

You will write the ratio test as a function, find a value of $k$, and count the pairs of a list that have infinitely many solutions.

The idea. Do not divide: cross-multiply. The lines cross when $a_1b_2 \neq a_2b_1$. If $a_1b_2 = a_2b_1$, they coincide when $a_1c_2 = a_2c_1$ and are parallel otherwise.

Your tasks.

1. In kind, return "one", "none" or "many" for the number of solutions of the pair.
2. In k_for_no_solution, return the $k$ for which $a_1x + b_1y + c_1 = 0$ and $kx + b_2y + c_2 = 0$ have no solution. Assume $b_1$ is not $0$.
3. Set many_count to how many pairs in pairs have infinitely many solutions.