SVM with a priori information about class probabilities
SVM doesn't take into account that prior knowledge about the distribution of the classes, so if you want a classifier that takes advantage of that, you'll need a different classifier.

In particular, if you know the distribution of each class and the priors (class balance), you can use Bayes rule to classify each point directly, without needing any fancy classifier.  This is the optimal (most accurate) way to classify, is not hard to implement, and doesn't require any training set.

If you know that each class has a bivariate normal distribution but you don't know its parameters (mean and covariance matrix), you can use the training set to estimate these parameters, then use Bayes rule.