Origin is a limiting point of a coaxial system of which

$x^2+y^2-6x-8y+1=0$is a member. The other limiting point, is

We know that the limit points other than the origin of the coaxial system of circles 

$x^2+y^2+2gx+2fy+c=0$ are given by

$\left(-\frac{gc}{g^2+f^2},-\frac{fc}{g^2+f^2}\right)$

Here, $g =-3, f=- 4, c =1.$

Hence, other limiting point is ( 3/25, 4/25).