conics

I graphed it here

https://www.desmos.com/calculator/ercszykueb

 

 

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.