Question

If \( \frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q} \) are in A.P., then:

• A. \( p, q, r \) are in A.P.
• B. \( p^2, q^2, r^2 \) are in A.P.
• C. \( \frac{1}{p}, \frac{1}{q}, \frac{1}{r} \) are in A.P.
• D. \( p + q + r \) are in A.P.

Hint:

For terms in A.P., the middle term should be the average of the first and third terms. Using algebraic manipulation, we can deduce the required sequence.

Answer 1

The Correct Option is B

Assuming \( \frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q} \) constitute an arithmetic progression (A.P.), the definition of an A.P. is applied: \[ 2 \times \frac{1}{r + p} = \frac{1}{q + r} + \frac{1}{p + q} \] The equation is then rearranged as follows: \[ \frac{1}{r + p} - \frac{1}{q + r} = \frac{1}{p + q} - \frac{1}{r + p} \] Upon cross-multiplication and simplification, the following is obtained: \[ q^2 - p^2 = r^2 - q^2 \] This implies that: \[ \Rightarrow p^2, q^2, r^2 \text{ are in A.P.} \] Therefore, the correct option is \( \boxed{(b)} \).