Geo4.82

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Find the equations to the transverse common tangents of the circles x^{2}+y^{2}=1,(x-1)^{2}+(y-3)^{2}=4\,.

Centres and radii of the given circles are

A(0,0),r_{1}=1,B(1,3),r_{2}=2\,

AB={\sqrt  {10}},r_{1}+r_{2}=3,AB>(r_{1}+r_{2})\,

Circles don't intersect with each other

Let the point C divides AB in the ratio 1:2

C=\left({\frac  {1.1+2.0}{3}},{\frac  {1.3+2.0}{3}}\right)\,

C=\left({\frac  {1}{3}},1\right)\,

Equation of the transverse tangents are

(x({\frac  {1}{3}})+y-1)^{2}=(x^{2}+y^{2}-1)({\frac  {1}{9}}+1-1)\,

(x+3y-3)^{2}=x^{2}+y^{2}-1\,

x^{2}+9y^{2}+9+6xy-18y-6x=x^{2}+y^{2}-1\,

8y^{2}+6xy-6x-18y+10=0\,

4y^{2}+3xy-3x-9y+5=0\,

Main Page:Geometry:Circles