As the main issue in pinching-antenna system design, antenna location optimization is key to realizing channel reconfigurability and system flexibility. Most existing works in this area adopt sophisticated optimization and learning tools to identify the optimal antenna locations in a numerical manner, where insightful understandings of the pinching antenna placement are still missing. Motivated by this research gap, this paper aims to carry out analytical optimization for pinching antenna placement, where closed-form solutions for the optimal antenna locations are obtained to reveal the impact of antenna placement on the system performance. In particular, for the user-fairness-oriented orthogonal multiple access (OMA) based transmission, analytical results are obtained to reveal that the pinching antenna needs to be activated at the place that would be beneficial to all served users; however, the users' distances to the waveguide have no impact on the location selection. For the greedy-allocation-based OMA transmission, an asymptotic study based on a high signal-to-noise ratio approximation is carried out to show that the optimal antenna location is in close proximity to the user who is nearest to the waveguide. For non-orthogonal multiple access (NOMA) based transmission, even with a user-fairness-oriented objective, the obtained analytical results show that the optimal antenna location is not the position that can benefit all users, but rather is near the user positioned closest to the waveguide.