If $x^2+y^2-2x-6y+9=0$and $x^2+y^2+6x-2y+1=0$are two circles, then

Given circles are

$x^2+y^2-2x-6y+9=0$

$x^2+y^2+6x-2y+1=0$

$C_1=(1,3)r_1=\sqrt{1+9-9}=1$

$C_2=(-3,1)r_2=\sqrt{9+1-1}=3$

$C_1{C}_2=\sqrt{{(1+3)}^2+{(3-1)}^2}$

$=\sqrt{16+4}$

$=\sqrt{20}$

$r_1+r_2=1+3=4$

$C_1{C}_2>r_1+r_2$

The given circle do not intersect

Number common tangents are 4

Internal centre of similitude $=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

$=(\frac{1(-3)+3\left(1\right)}{1+3},\frac{1\left(1\right)+3\left(3\right)}{1+3})$

$=(\frac{-3+3}{4},\frac{1+9}{4})$

$=(0,\frac{5}{2})$

External centre of similitude $=(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n})$

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

$=(\frac{-6}{-2},\frac{-8}{-2})$

$=(3,4)$