A

Step 1: The power of a lens is given by the formula \( P = \frac{1}{f} \), where \( P \) is the power in diopters and \( f \) is the focal length in meters.
Step 2: For two lenses in contact, the total power is given by \( P_{total} = P_1 + P_2 \). Here, \( P_{total} = +10 \) diopters.
Step 3: For two lenses separated by a distance 'd', the total power can be expressed as \( P_{total} = P_1 + P_2 - \frac{d}{f_1 f_2} \), where \( d = 0.25 m \).
Step 4: According to the question, when the lenses are 0.25 m apart,\( P_{total} = +6 \) diopters.
Step 5: From the two equations, we have:
1. \( P_1 + P_2 = +10 \)
2. \( P_1 + P_2 - \frac{0.25}{f_1 f_2} = +6 \)
Step 6: Rearranging the second equation gives \( \frac{0.25}{f_1 f_2} = +4 \). Thus, we have \( f_1 f_2 = \frac{0.25}{4} = 0.0625 \).
Step 7: Let \( P_1 = x \) and \( P_2 = 10 - x \). From the relationship of the focal lengths, we know that:
\( f_1 = \frac{1}{x} \) and \( f_2 = \frac{1}{10-x} \).
Step 8: Substitute in the product equation: \( \frac{1}{x(10 - x)} = 0.0625 \). Thus, \( x(10 - x) = 16 \).
Step 9: This simplifies to the quadratic equation: \( x^2 - 10x + 16 = 0 \).
Step 10: Solving this quadratic equation, we get the focal lengths as \( f_1 \) and \( f_2\) as approximately 0.4 m and 0.25 m (or vice versa).
Therefore, the focal lengths of the lenses are 0.4 m and 0.25 m.