This paper proposes fast, reliable, and minimal non-iterative relative pose solvers under planar motion constraint. Relative pose estimation is popularly utilized in many important problems such as visual odometry and SLAM, and planar motion is common for mobile robots and vehicles on floors and roads. We transform the original problem formulation of finding intersections of two ellipses into more accessible form of finding intersections of a line and unit circle. Such transformation leads to a non-iterative and closed-form solver, which enables significant speed-up compared to previous methods. The proposed algorithm is almost 9 times faster than the previous minimal solver with planar motion and around 90 times faster than the previous minimal solver with general motion. In addition, our algorithms provide reliable relative pose in degeneracy of the previous minimal planar solvers. Effectiveness of the proposed algorithms is demonstrated with two types of experiments: relative pose estimation with synthetic data and monocular visual odometry with real image sequences.