Fairness and privacy are two vital pillars of trustworthy machine learning. Despite extensive research on these individual topics, the relationship between fairness and privacy has garnered significantly less attention. In this paper, we investigate the relationship between fairness, privacy, and accuracy using the information theoretic concept of Chernoff Information. We define Chernoff Difference, a tool that allows us to analyze the relationship between fairness, privacy, and accuracy. We then show that for Gaussian distributions, this value behaves in 3 distinct ways (depending on the distribution of the data). We highlight the distributions that these cases entail as well as their fairness and privacy implications. Additionally, we show that Chernoff difference acts as a proxy to the steepness of the fairness-accuracy curves. This work provides a foundation towards a more comprehensive understanding of the relationship between fairness, privacy and accuracy in machine learning and motivates the use of Chernoff information in this domain.