A chord CD is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

This is a complex geometry problem requiring advanced circle theorems and algebraic manipulation. A chord of length equal to radius creates a 60° central angle (equilateral triangle property). Two circles inscribed on each side of the chord, each touching the chord at its midpoint and the original circle, creates a configuration where the ratio of their areas involves √3 terms. The answer A (97 + 56√3) results from this geometric analysis, though the full derivation is lengthy.