The limiting points of the coaxial system containing the two circles 

$x^2+y^2+2x-2y+2=0$ and $25\left(x^2+y^2\right)-10x-80y+65=0$ are 

The equation of the radical axis of the given circles is $4x+2y-1=0$

Therefore, the equation of family of coaxial circles is 

$\begin{array}{l} x^2+y^2+2x-2y+2+\lambda (4x+2y-1)=0\\ \Rightarrow x^2+y^2+2x(1+2\lambda )+2y(\lambda -1)+2-\lambda =0\end{array}$

The coordinates of the centre and radius of this circle are 

$(-(2\lambda +1),-(\lambda -1\left)\right)$and $\sqrt{(2\lambda +1{)}^2+(\lambda -1{)}^2-2}$ 

For limiting points, we must have 

Radius =0 

$\Rightarrow (2\lambda +1{)}^2+(\lambda -1{)}^2+\lambda -2=0\Rightarrow \lambda =0,-3/5$

Hence, the limiting points are (-1, 1) and (1/5 , 8/5)