Geo4.82

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

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

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

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

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

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