I have a problem regarding simultaneous equation solving. The question gives the equation of a straight line 'y + kx = -2' and also the equation of a hyperbola (I don't have the exact equation written down but it was something like 'y(k-8)=x'). The straight line represents a footpath going between two circular pools. The purpose is to find 'appropriate values of k' so that the path (straight line) does not intercept the pools (lines of the hyperbola). I use the following quadratic formula:

I remember something about using the discriminant (b^2-4ac) and making it equal less than 0 to show there are no real roots/answers. I just can't remember what else to do. Could someone please show me full steps to solve a problem like this?

What I have figured out how to do so far:

The second part of the question was to also show the possible 'range of values' for k.

Thank-you to anyone who can assist. I really appreciate it.