Question: The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units. What are the values of r and R?<br>Statement-I: r is an integer.<br>Statement-II: R is an integer.

Consider the following for the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions : Choose option (a) If the Question can be answered by one of the Statements alone, but not by the other. (b) If the Question can be answered by either Statement alone. (c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone. (d) If the Question cannot be answered even by using both Statements together.

  1. A. If the Question can be answered by one of the Statements alone, but not by the other.
  2. B. If the Question can be answered by either Statement alone.
  3. C. If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. If the Question cannot be answered even by using both Statements together. ✓

Correct Answer: D. If the Question cannot be answered even by using both Statements together.

Explanation

The chord of the outer circle acts as a tangent to the inner circle. By the Pythagorean theorem, R^2 - r^2 = (24/2)^2 = 144. Using both statements, we know R and r are integers. We can factor this as (R-r)(R+r) = 144. Setting up factor pairs of 144 with the same parity yields multiple valid solutions: for example, (2, 72) gives R=37, r=35; (6, 24) gives R=15, r=9; (8, 18) gives R=13, r=5. Because there are multiple distinct integer solutions, we cannot uniquely determine the exact values of r and R even when using both statements together.

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