Question

If \[ \frac{1}{q + r}, \quad \frac{1}{r + p}, \quad \frac{1}{p + q} \] {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:

When terms are in arithmetic progression, apply the standard condition \( 2b = a + c \) to find relations between terms.

Answer 1

The Correct Option is B

Step 1: We know that if \( \frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q} \) are in arithmetic progression, then the condition for an arithmetic progression is: \[ 2 \cdot \left( \frac{1}{r + p} \right) = \frac{1}{q + r} + \frac{1}{p + q}. \] Step 2: Simplifying the equation, we get a relation between \( p, q, r \). After solving this, it turns out that the squares of \( p, q, r \) satisfy an arithmetic progression. Therefore, the correct answer is \( p^2, q^2, r^2 \) are in A.P.