Question

`x+-y = 0` are asymptotes to the family of Rectangular Hyperbolas (RH) `x^2-y^2=+-c^2`. How do you prove that the multitude of straight lines `x+-y=a` are asymptotes to the RH family?

Answer 1

Answer, a mon avis. Perhaps, this could possibly be disproved, in another answer. Please avoid editing my answer.

Explanation:

If `( x, y ) = ( r, theta)` is any point on any straight line

`x +- y = a`, and any RH `x^2 - y^2 = +- c^2`,

`in Q_1, Q_2, Q_3, Q_4`,

as `r to oo, theta to pi/4, (3pi)/4, -(3pi)/4, - pi/4`

respectively.

https://www.reddit.com/r/askscience/comments/512ts7/why_do_parallel_lines_meet_at_infinity/

See graphs for convergence to point at `oo`, in the respective

direction, as we advance for higher r.

Graph near origin, for Q_1 ( similar graphs can be created for other

quadrants):
graph{((x^2-y^2)^2-1)((x^2-y^2)^2-9)(x-y-1)(x-y+1)(x-y-4)(x-y+4)(x-y)=0}

Graph for r > 10000:

graph{(x^2-y^2-1)(x^2-y^2-9)(x-y-1)(x-y+1)(x-y-4)(x-y+4)(x-y)=0[0 20000 0 10000]}

Observe that scaling produces alignment, upon marching to `oo`, in

the direction `theta = pi/4`.