Geo4.67

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Find the angle between the pair of tangents drawn from (1,3) to the circles x^{2}+y^{2}-2x+4y-11=0\,

The equation of the pair of tangents is

[S_{1}]^{2}=SS_{{11}}\,

From the given circle, radius is {\sqrt  {1+4+11}}=4\,

S_{1}=x(1)+y(3)-1(x+1)+2(y+3)-11=5y+5\,

S_{{11}}=1^{2}+3^{2}-2(1)+4(3)-11=9\,

Now the equation is

25(y+1)^{2}=(x^{2}+y^{2}-2x+4y-11)9\,

25y^{2}+25+50y=9x^{2}+9y^{2}-18x+36y-99\,

-9x^{2}+16y^{2}+18x-36y+124=0\,

\tan \theta ={\frac  {2\cdot 4({\sqrt  {9}})}{16-9}}\,

\tan \theta ={\frac  {24}{7}}\,

Main Page:Geometry:Circles