Geo5.2.27

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Find the pole of the line 3x-5y-9=0\, w.r.t the ellipse 4x^{2}+8y^{2}-16x+15=0\,

Let the pole of the given line w.r.t the given ellipse be P(x_{1},y_{1})\,

Equation of polar of P is

4xx_{1}+8yy_{1}-8(x+x_{1})+15=0\,

x(4x_{1}-8)+8yy_{1}+15-8x_{1}=0\,

Comparing this equation with 3x-5y-9=0\,

{\frac  {4x_{1}-8}{3}}={\frac  {8y_{1}}{-5}}={\frac  {15-8x_{1}}{-9}}\,

{\frac  {4x_{1}-8}{3}}={\frac  {15-8x_{1}}{-9}},{\frac  {8y_{1}}{-5}}={\frac  {15-8x_{1}}{-9}}\,

-36x_{1}+72=45-24x_{1},-72y_{1}=-75+40x_{1}\,

x_{1}={\frac  {27}{12}}={\frac  {9}{4}},-72y_{1}=-75+90,y_{1}={\frac  {5}{-24}}\,

Hence the pole of the given line is \left({\frac  {9}{4}},{\frac  {-5}{24}}\right)\,


Main Page:Geometry:The Ellipse