  The present paper investigates the binary system of quasars in the framework
of the Circular Restricted Three-Body Problem. The parametric evolution of
libration points, the geometry of zero-velocity curves are one of the crucial
aspects of our study. The multivariate form of NR method is applied to study
the basin of attraction connected with libration points. The algorithm for
using the Newton-Raphson method is slightly modified in order to avoid the
unnecessary delay in the convergence of initial conditions. The impact of
parameters on the shape of the basin of attraction and the number of iterations
needed for the convergence of initial conditions are explored. We carry out an
exhaustive (numerical) study to show the influence of these parameters on
converging regions in basins of convergence. We unveil the existence of fractal
structure in the basin of attraction using the method of basin entropy. In
almost all cases, the existence of fractal structure is found throughout the
basins of attraction.
