Geo5.3.21

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Show that the locus of the pole of any tangent to the circle x^{2}+y^{2}=a^{2}\, w.r.t the hyperbola x^{2}-y^{2}=a^{2}\, is the circle itself.

The equation to the polar of P(x_{1},y_{1})\, w.r.t the hyperbola is

{\frac  {xx_{1}}{a^{2}}}-{\frac  {yy_{1}}{a^{2}}}=1\,

If this is a tangent to the circle x^{2}+y^{2}=a^{2}\, if

{\frac  {|0-0-1|}{{\sqrt  {{\frac  {x_{1}^{{2}}}{a^{2}}}+{\frac  {y_{1}^{{2}}}{a^{2}}}}}}}=a\,

1=a^{2}{\frac  {(x_{1}^{{2}}+y_{1}^{{2}})}{a^{2}}}\,

x_{1}^{2}+y_{1}^{2}=1\,

Therefore,the locus of P is x^{2}+y^{2}=1\, which is a circle.


Main Page:Geometry:Hyperbola