We consider a generalization of the classical planar maximum coverage location problem (PMCLP) in which partial coverage is allowed, facilities have adjustable quality of service (QoS) or service range, and demand zones and service zone of each facility are represented by two-dimensional spatial objects such as rectangles, circles, polygons, etc. We denote this generalization by PMCLP-PC-QoS. A key challenge in this problem is to simultaneously decide position of the facilities on a continuous two-dimensional plane and their QoS. We present a greedy algorithm and a pseudo-greedy algorithm for it, and provide approximation ratios for these approaches. We also investigate theoretical properties and propose exact algorithms for solving: (1) PMCLP-PC-QoS where demand and service zones are represented by axis-parallel rectangles (denoted by PMCLP-PCR-QoS), which also has applications in camera surveillance and satellite imaging; and (2) one dimensional PMCLP-PC-QoS which has applications in river cleanups. These results extend and strengthen the only known exact algorithm for PMCLP-PCR-QoS with fixed and same QoS by Bansal and Kianfar [INFORMS Journal on Computing 29(1), 152-169, 2017]. We present results of our computational experiments conducted to evaluate the performance of our proposed exact and approximation algorithms.