How do you write polar equations of hyperbolas, from the polar equations of their asymptotes?

The polar equation of a straight line is

#r cos (theta - alpha ) = p >= 0,#

#theta in ( alpha - pi/2, alpha + pi/2)#, with #( p, alpha )# as the

foot of the perpendicular from the pole r = 0, on the straight line.

So, the polar equation of hyperbolas having

#r cos (theta - alpha ) = a# and

#r cos (theta - beta ) = b#

as asymptotes is

#(r cos (theta - alpha ) - a)(r cos (theta - beta ) - b) = C#

Note that #alpha, beta, a, b and C# are parameters.

For example, consider the asymptotes #x+-y = 1#.

Their polar equations are #r (cos theta +- sin theta ) = 1#.

So, the polar equations of the related hyperbolas are

#(r (cos theta + sin theta ) - 1)(r (cos theta - sin theta ) - 1) = C#.

Upon setting C = 1, this gives the hyperbola

#r(cos^2theta - sin^2theta)- 2 cos theta = 0#

that has the Cartesian form

#x^2 - y^2 - 2 x = 0#,

and this in the standard form is

#( x - 1 )^2 - y^2 = 1#

Graph for the illustrative hyperbola

#(r (cos theta + sin theta ) - 1)(r (cos theta - sin theta ) - 1) = 1#.
graph{((x-1)^2-y^2-1)((x-1)^2-y^2)=0[-2 6 -2 2]}