Geo5.2.29

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Show that the two lines 9x+2y=1,3x+2y=11\, are conjugate w.r.t the ellipse 3x^{2}+2y^{2}=1\,

Let P(x_{1},y_{1}),Q(x_{2},y_{2})\, be the poles of the two given lines w.r.t the ellipse.

The equation of polar of P is

3xx_{1}+2yy_{1}=1\,

Comparing this with 9x+2y=1\, we get

{\frac  {3x_{1}}{9}}={\frac  {2y_{1}}{2}}={\frac  {1}{1}}\,

(3,1)\,

The equation of polar of Q is

3xx_{2}+2yy_{2}=1\,

Comparing this with 3x+2y=11\, we get

{\frac  {3x_{2}}{3}}={\frac  {2y_{2}}{2}}={\frac  {1}{11}}\,

\left({\frac  {1}{11}},{\frac  {1}{11}}\right)\,

The condition for the two lines to be conjugate is S_{{12}}=0\,

3(3)({\frac  {1}{11}})+2(1)({\frac  {1}{11}})-1={\frac  {9}{11}}+{\frac  {2}{11}}-1={\frac  {11}{11}}-1=0\,

Hence the two lines are conjugate.


Main Page:Geometry:The Ellipse