The group law on a hyperbola

And as described in the previous post, it's possible to define a group law for points on a circle and it's easy to generalize the geometric interpretation of this operation to points on other conics. Here's an example - of what adding the points (5/4,3/4) and (-5/4,-3/4) to get the point (-17/8,-15/8) on the hyperbola x² -y² = 1 looks like, using the point O = (1,0) as the additive identity, where we find the slope of the line through the two points we want to add, find the second point where the line through O with that slope intersects the curve and call that point the sum: