  We study the number of limit cycles and the bifurcation diagram in the
Poincar\'{e} sphere of a one-parameter family of planar differential equations
of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in
previous papers. We prove that there is a value $b^*>0$ such that the limit
cycle exists only when $b\in(0,b^*)$ and that it is unique and hyperbolic by
using a rational Dulac function. Moreover we provide an interval of length
27/1000 where $b^*$ lies. As far as we know the tools used to determine this
interval are new and are based on the construction of algebraic curves without
contact for the flow of the differential equation. These curves are obtained
using analytic information about the separatrices of the infinite critical
points of the vector field. To prove that the Bendixson-Dulac Theorem works we
develop a method for studying whether one-parameter families of polynomials in
two variables do not vanish based on the computation of the so called double
discriminant.
