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Show that the points of intersection of the asymptotes of hyperbola with its directrices lie on the auxiliary circle.

Let the hyperbola be {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\,

Equations of the asymptotes of the hyperbola are

{\frac  {x}{a}}-{\frac  {y}{b}}=0,{\frac  {x}{a}}+{\frac  {y}{b}}=0\,

Equations to the directrices are x=\pm {\frac  {a}{e}}\,

Solving the equations we get the points of intersection are ({\frac  {a}{e}},\pm {\frac  {b}{e}})\,

Substituting in the equation of the auxiliary circle x^{2}+y^{2}=a^{2}\,

We have {\frac  {a^{2}}{e^{2}}}+{\frac  {b^{2}}{e^{2}}}=a^{2}\,

{\frac  {a^{2}+b^{2}}{e^{2}}}=a^{2},{\frac  {a^{2}e^{2}}{e^{2}}}=a^{2},a^{2}=a^{2}\,

Hence the points lie on the auxiliary circle.

Main Page:Geometry:Hyperbola