Geo5.2.30

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Find the value of k,if the linesx+y+k=0,3x-2y-7=0\, are conjugate w.r.t the ellipse x^{2}+3y^{2}=9\,

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

Equation of the polar of the first line is

xx_{1}+3yy_{1}=9\,

Comparing this with the first line,

{\frac  {x_{1}}{1}}={\frac  {3y_{1}}{1}}={\frac  {-9}{k}}\,

x_{1}={\frac  {-9}{k}},y_{1}={\frac  {-3}{k}}\,

Equation of the polar of Q is

xx_{2}+3yy_{2}-9=0\,

Comparing this with the second line,we get

{\frac  {x_{2}}{3}}={\frac  {3y_{2}}{-2}}={\frac  {9}{7}}\,

x_{2}={\frac  {27}{7}},y_{2}={\frac  {-6}{7}}\,

Now the condition for the two lines to be conjugate is x_{1}x_{2}+3y_{1}y_{2}-9=0\,

({\frac  {-9}{k}})({\frac  {27}{7}})+3({\frac  {-3}{k}})({\frac  {-6}{7}})-9=0\,

-243+54-63k=0,-63k=189,k=-3\,


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