Geo4.9

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Find the power of the point (2,-1) with respect to 3x^{2}+3y^{2}+4x+2y+6=0\,

Simplifying the equation,

x^{2}+y^{2}+{\frac  {4}{3}}x+{\frac  {2}{3}}y+2=0\,

Let the point be P and centre be C and radius r

Power of the point is

[CP]^{2}-r^{2}\,

Centre of the given circle is

\left({\frac  {-2}{3}},{\frac  {-1}{3}}\right)\,

Now

[CP]^{2}=(2+{\frac  {2}{3}})^{2}+(-1+{\frac  {1}{3}})^{2})={\frac  {64}{9}}+{\frac  {4}{9}}\,

[CP]^{2}={\frac  {68}{9}}\,

r^{2}={\frac  {4}{9}}+{\frac  {1}{9}}-2={\frac  {-13}{9}}\,

Hence the power of the point is

[CP]^{2}-r^{2}={\frac  {68}{9}}+{\frac  {13}{9}}\,

{\frac  {81}{9}}=9\,

The power of the point is 9.

Main Page:Geometry:Circles