The equation of the circle passing through (0, 0) and belonging to the system of circles of which (3, 1) and (-1, 5) are limiting points, is

Circles having( 3, 1) and (-1, 5) as limiting points
are $S_1\equiv (x-3{)}^2+(y-1{)}^2=0$ and $S_2\equiv (x+1{)}^2+(y-5{)}^2=0$

The equation of the family of circles is

$\begin{array}{l}{S}_1+\lambda \left(S_1-S_2\right)=0\\ \Rightarrow (x-3{)}^2+(y-1{)}^2+\lambda (-8x+8y-16)=0\end{array}$                              ... (i)

It passes through (0, 0).

$\therefore 10-16\lambda =0\Rightarrow \lambda =\frac{5}{8}$

Substituting the value of $\lambda$ in (i), we get

$x^2+y^2-11x+3y=0$ as the equation of the required circle.