Geo5.3.23

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If the polar of a point w.r.t ellipse S=0\, touch the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\,,then show that the locus of point is is the hyperbola.

The polar of point P w.r.t the ellipse is {\frac  {xx_{1}}{a^{2}}}+{\frac  {yy_{1}}{b^{2}}}=1\,

From this line l={\frac  {x_{1}}{a^{2}}},m={\frac  {y_{1}}{b^{2}}},n=-1\,

The condition for this line to touch the hyperbola is

a^{2}l^{2}-b^{2}m^{2}=n^{2}\,

a^{2}({\frac  {x_{1}^{{2}}}{a^{4}}})-b^{2}({\frac  {y_{1}^{{2}}}{b^{4}}}=(-1)^{2}\,

{\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}=1\,

Hence the locus of P is {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, which is a hyperbola.


Main Page:Geometry:Hyperbola