This paper focuses on efficiently evaluating derivatives of the 2-D collision probability, treated as a function of parameters, which appear as linear forms in the entries of the covariance matrix. This boils down to computing moments of the associated Gaussian measure restricted to a disk. Specifically, we propose an optimization-based solution to computing the maximum collision probability when the covariance data is unreliable, implementing an alternative method to the traditional k-scaled covariance approach. Preliminary results indicate our method's potential for improving the understanding of Pc's validity as a measure of conjunction likelihood.