I graphed it here
I let m=tan(t) where t is the angle that is made with the positive x axis (for any line with a gradient of m)
It is the locus of points that make the intersection of the solid red and green lines that is relevant.
t is in radians
You can slide the value of t to see what happens to this intersection.
The locus is definitely a hyperbola (just by visual inspection)
Since it is a hyperbola the eccentricity has to be more than 1 so I am going for 3
Of course you were meant to nut it out without graphing. You could probably do this just by thinking about a few points.
My way is more fun and much more time consuming.