All chords through an external point to the circle $x^{2}+y^{2}=16$ are drawn having length $\ell$ which is a positive integer. The sum of the squares of the distances from the centre of circle to these chords is

For any chord of integer length l, the distance d from the center of the circle x^2 + y^2 = 16 (radius R = 4) satisfies d^2 = 16 - l^2/4. Since the chords pass through an external point, for each integer length l from 1 to 7, there are exactly two such chords (tangents to the concentric circle of radius d), while for l = 8 (the diameter), there is only one chord at distance d = 0. Summing the squares of these distances, we get 2 * Sum_{l=1}^{7} (16 - l^2/4) = 224 - 70 = 154.