This paper addresses one of the classical problems in random matrix theory-- finding the distribution of the maximum eigenvalue of the correlated Wishart unitary ensemble. In particular, we derive a new exact expression for the cumulative distribution function (c.d.f.) of the maximum eigenvalue of a $2\times 2$ correlated non-central Wishart matrix with rank-$1$ mean. By using this new result, we derive an exact analytical expression for the outage probability of $2\times 2$ multiple-input multiple-output maximum-ratio-combining (MIMO-MRC) in Rician fading with transmit correlation and a strong line-of-sight (LoS) component (rank-$1$ channel mean). We also show that the outage performance is affected by the relative alignment of the eigen-spaces of the mean and correlation matrices. In general, when the LoS path aligns with the least eigenvector of the correlation matrix, in the {\it high} transmit signal-to-noise ratio (SNR) regime, the outage gradually improves with the increasing correlation. Moreover, we show that as $K$ (Rician factor) grows large, the outage event can be approximately characterized by the c.d.f. of a certain Gaussian random variable.