Geo5.2.25

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Find the pole of the line x-y+2=0\, with w.r.t the ellipse x^{2}+2y^{2}-4x+12y+14=0\,

Le the pole of the given line to the given ellipse be P(x_{1},y_{1})\,

Equation of the polar of P is with respect to the ellipse is xx_{1}+2yy_{1}-2(x+x_{1})+6(y+y_{1})+14=0\,

x(x_{1}-2)+y(2y_{1}+6)+6y_{1}-2x_{1}+14=0\,

Comparing this equation to the line is

{\frac  {x_{1}-2}{1}}={\frac  {2y_{1}+6}{-1}}={\frac  {6y_{1}-2x_{1}+14}{2}}\,

{\frac  {x_{1}-2}{1}}={\frac  {2y_{1}+6}{-1}},x_{1}+2y_{1}=-4\,. let this equation be 1.

{\frac  {x_{1}-2}{1}}={\frac  {6y_{1}-2x_{1}+14}{2}},2x_{1}-3y_{1}=9\,

Solving these two, we get

\left({\frac  {6}{7}},{\frac  {-17}{7}}\right)\,


Main Page:Geometry:The Ellipse