Circles \mathcal{P} and \mathcal{Q} have radii 1 and 4 , respectively, and are externally tangent at point A. Point B is on \mathcal{P} and point C is on \mathcal{Q} so that line B C is a common external tangent of the two circles. A line \ell through A intersects \mathcal{P} again at D and intersects \mathcal{Q} again at E. Points B and C lie on the same side of \ell, and the areas of \triangle D B A and \triangle A C E are equal. This common area is \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.