We study hyper-elliptic Seiberg-Witten curves for N = 2 pure super Yang-Mills theories. We study the locus of vanishing discriminant of the curve and compute charges for all 2r + 1 and 2r + 2 massless dyons for Sp(2r) and SU(r + 1) with rank r > 1 in a region of moduli space. We apply a double-discriminant method to locate Argyres- Douglas singularity loci in the moduli space, by applying another discriminant operator, with respect to another variable to be one of the gauge-invariant moduli u. Two BPS dyons become massless where two different solutions of vanishing discriminant intersect. This complex codimension-2 locus in the moduli space is easily captured by demanding an extra condition of vanishing double discriminant, whose order of vanishing also determines whether dyons are Argyres-Douglas type or not. The intersection locus in the moduli space is cusp-like and node-like for mutually non-local and local pairs of BPS dyons respectively.