Auxiliary Circle and Director Circle of a Hyperbola

As with the ellipse, two circles are naturally associated with the hyperbola:

• The auxiliary circle, on the transverse axis as diameter: $x^2+y^2=a^2$ (used for the parametric form, section 7.3.3).
• The director circle, the locus of perpendicular-tangent intersections (section 7.3.7): $x^2+y^2=a^2-b^2$, real only when $a>b$.

Worked example: $9x^2-16y^2=144$. Rewriting, $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$, so $a^2=16,\,b^2=9$ (and indeed $a^2>b^2$, so the director circle is real).

• Auxiliary circle: $x^2+y^2=a^2=16$.
• Director circle: $x^2+y^2=a^2-b^2=16-9=7$. …