Minkowski spacetime diagrams, what does the hyperbola represent?

[knowlewj01]

Homework Statement

Draw a clearly labelled “Minkowski spacetime” diagram illustrating two events
((1) a farmer firing his laser gun at his cow, which is sitting along his positive x-direction, and
(2) the cow dying) as observed by two observers (S at rest in the farmer’s and cow’s frame,
and S’ moving at 0.5 c in the direction of the negative x direction.
Show on the diagram, qualitatively, hyperbolae which determine the scale of the ct, x and
ct’, x’ axes. Using the diagram, is there any frame S’ where it appears that the cow dying
caused the farmer to shoot his gun?

Homework Equations

See attached file for my diagram so far

The Attempt at a Solution

on my diagram you can see the S and S' frames, where S' is moving in the negative x direction with a velocity of 0.5c. I'm aware that the angle between the axes of the two different frames are governed by the velocity of the S' frame. The question asks for hyperbolae which determine the scale, I'm not sure where these should go. The black dot shows the farmer firing the gun and the red dot show the time when the cow dies in each frame, should the hyperbola pass through both of these points?

 

[knowlewj01]

changed my mind, the gun is a laser so information about the cow's death travels at the speed of light, the dotted line shows lightlike region, the region where the event of the cow's death occurs. The cow is a certain distance from the farmer, so in the S frame the sequence of events happens as in the vertical and horizontal axes. In the S' frame the events appear as in the axes at an angle to S frame. In both frames the events are on the lightlike region. I still cannot see the parts of the diagram to draw the hyperbolae whos tangents bive the x to x' and ct to ct' transformations. Anyone know where to draw the hyperbolae?