Consider the following statements: 1. If (a+b) is directly proportional to (a-b), then (a^{2}+b^{2}) is directly proportional to ab. 2. If a is directly proportional to b, then (a^{2}-b^{2}) is directly proportional to ab. Which of the statements given above is/are <strong>CORRECT</strong>?

  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2 ✓
  4. D. Neither 1 nor 2

Correct Answer: C. Both 1 and 2

Explanation

If a+b = k(a-b), then solving for a gives a = \frac{k+1}{k-1}b, meaning a \propto b. If a=mb, then a^2+b^2 = b^2(m^2+1) and ab = mb^2, making them proportional. Thus statement 1 is correct. Similarly, if a=mb, a^2-b^2 = b^2(m^2-1) and ab = mb^2, so they are proportional. Thus statement 2 is correct.

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