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Two circles, C₁ with equation x² + y² = 9 and C₂ with equation (x - 3)² + (y - 4)² = 16, intersect at points X and Y. A third circle, C₃ with equation (x - h)² + (y - k)² = r², satisfies the following conditions: (i) The centers of C₁, C₂, and C₃ are collinear. (ii) C₁ and C₂ are completely enclosed within C₃. (iii) C₃ is tangent to C₁ at point M and to C₂ at point N. The line through X and Y intersects C₃ at points Z and W, and a common tangent to C₁ and C₃ is also a tangent to the parabola x² = 8xy. Below are expressions in List-I and their corresponding values in List-II: LIST-I: (I) 2h + k (II) Length of ZW divided by length of XY (III) Ratio of the area of triangle MZN to the area of triangle ZMW (IV) α LIST-II: (P) 6 (Q) √6 (R) 5/4 (S) 21/5 (T) 2√6 (U) 10/3 Which of the following is the only mismatched pair?

1. (1) (I) and (V) with (U)
2. (2) (I) with (P)
3. (3) (III) with (R)
4. (4) (III) with (S)

Correct answer: (1) (I) and (V) with (U)

Solution

The mismatch occurs because the pair (I) and (V) with (U) does not align with the given conditions and calculations for the circle and tangent properties.

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