A magnetoelectrostatic trap consists of a spindle cusp magnetic field configuration and transverse electrostatic potentials applied to the electrodes at the ring and point cusp ends. Plasma confinement studies in this trap is made with a single positive charged particle. When the plasma is injected in such a system a potential well for the positive ions is formed where they could be trapped. Three-dimensional trajectory of the particle was developed by a single-gap mathematical model. The trajectory of a positive ion in the system was analyzed numerically by Runge-Kutta fourth order method. The equations of motion were constructed from the Hamiltonian function. The three-dimensional trajectory of an ion was traced for various possible input parameters. When the electrostatic potential and angle of injection are kept zero almost all the injected ions (post-cusp region) escape through the aperture. But there exists a critical injection velocity and magnetic field intensity to reflect the particle at axial cusp region. In this forward and backward motion the particle moves in a helical path encircling about the same magnetic lines of force. While introducing some potential difference to the electrodes in the cusp ends, the particle exhibits a "double helix" trajectory about the axis of the spindle cusp configuration. This double helical path leads long duration confinement of the particle in such a system and obviously cusp losses are suppressed. The results are reported for a wide range of input parameters.