#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?

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Answer 1 · 2018-07-02T03:14:07

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

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.

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#.