Indeed,
This can only be a tangent when the equation has exactly one solution, that is, like you stated when the discriminant of is equal to zero.
Hence you must find k such that
So yeah, I'm looking at C1 past papers for my exam tomorrow, and question 10 part iii to this paper has confused the hell out of me. I understand that you're meant to equate the discriminant of the quadratic to 0, but how do you know that there's only one solution? It's just hurting my brain =/.