Geo4.84

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Write down the equation of the common tangent if the two circles x^{2}+y^{2}+6x-6y+2=0,x^{2}+y^{2}-2x=0toucheachother\,

The centre and radii of the given circles

A(-3,3),r_{1}={\sqrt  {9+9-2}}=4,B(1,0),r_{2}={\sqrt  {1}}=1\,

The point of contact divides the the two circles in the ratio 4:1

The point is

\left({\frac  {4\cdot 1+1\cdot (-3)}{4+1}},{\frac  {4\cdot 0+1\cdot 3}{5}}\right)\,

\left({\frac  {1}{5}},{\frac  {3}{5}}\right)\,

the equation of common tangent w.r.t the circle is

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

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

S_{1}={\frac  {16x-12y+4}{5}}\,

S_{1}={\frac  {16x-12y+4}{5}}\,

S_{{11}}={\frac  {1}{25}}+{\frac  {9}{25}}+{\frac  {6}{5}}-{\frac  {18}{5}}+2=0\,

Now the equation of the common tangent is

(16x-12y+4)^{2}=25(x^{2}+y^{2}+6x-6y+2)(0)\,

4x-3y+1=0\,

Main Page:Geometry:Circles