The equation of the transverse common tangent of the circles $x^2+y^2-6x-8y+9=0$ and $x^2+y^2+2x-2y+1=0$ is

Let

$S_1:x^2+y^2-6x-8y+9=0$

Here. centre is $C_1\left(3,4\right)$ and radius $r_1=4$ units.

$S_2:x^2+y^2+2x-2y+1=0$

Here. centre is $C_2\left(-1,1\right)$ and radius is $r_2=1$ unit.

$C_1{C}_2=\sqrt{16+9}=5=r_1+r_2$

Therefore, given circles are non-intersecting and touces each other.

Required equation of transverse common tangent of the circles

$S_1-S_2=0$

$\Rightarrow \left(x^2+y^2-6x-8y+9\right)-\left(x^2+y^2+2x-2y+1\right)=0$

$\Rightarrow 4x+3y-4=0$