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The polar of any point on the ellipse S=0\, w.r.t the hyperbola {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, will touch the ellipse.

Equation of polar of P(x_{1},y_{1})\, w.r.t the given hyperbola is

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

This will touch the given ellipse if


From the equation of polar l={\frac  {x_{1}}{a^{2}}},m={\frac  {-y_{1}}{b^{2}}},n=-1\,

Substituting these values in the condition of tangency,we get

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\,

Therefore,the point P(x_{1},y_{1})\, lies on the ellipse.

Hence the polar touches the ellipse.

Main Page:Geometry:Hyperbola