If `a ,b ,a n dc` are in A.P. `p ,q ,a n dr` are in H.P., and `a p ,b q ,a n dc r` are in G.P., then `p/r+r/p` is equal to `a/c-c/a` b. `a/c+c/a` c. `b/q+q/b` d. `b/q-q/b`

Options

A. `a/c-c/a`
B. `a/c+c/a`
C. `b/q+q/b`
D. `b/q-q/b`

Explanation

To solve the problem step by step, we need to analyze the given conditions carefully.

### Step 1: Understand the conditions

1. **A.P. Condition**: If \( a, b, c \) are in Arithmetic Progression (A.P.), then:
\[ b = \frac{a + c}{2} \]

2. **H.P. Condition**: If \( p, q, r \) are in Harmonic Progression (H.P.), then:
\[ q = \frac{2pr}{p + r} \]

3. **G.P. Condition**: If \( ap, bq, cr \) are in Geometric Progression (G.P.), then:
\[ bq^2 = ap \cdot cr \]

### Step 2: Find \( \frac{p}{r} + \frac{r}{p} \)

We need to find:
\[ \frac{p}{r} + \frac{r}{p} = \frac{p^2 + r^2}{pr} \]

### Step 3: Substitute \( q \) from H.P. into the G.P. condition

From the H.P. condition, we have:
\[ q = \frac{2pr}{p + r} \]
Substituting this into the G.P. condition:
\[ b \left( \frac{2pr}{p + r} \right)^2 = ap \cdot cr \]
Expanding this gives:
\[ b \cdot \frac{4p^2r^2}{(p + r)^2} = ap \cdot cr \]

### Step 4: Rearranging the equation

Rearranging the equation, we have:
\[ 4bp^2r^2 = ap \cdot cr \cdot (p + r)^2 \]

### Step 5: Finding \( \frac{p^2 + r^2}{pr} \)

Now, let's express \( \frac{p^2 + r^2}{pr} \):
\[ \frac{p^2 + r^2}{pr} = \frac{(p + r)^2 - 2pr}{pr} = \frac{(p + r)^2}{pr} - 2 \]

### Step 6: Substitute \( p + r \)

From the H.P. condition:
\[ p + r = \frac{2pr}{q} \]
Substituting this into our expression:
\[ \frac{\left( \frac{2pr}{q} \right)^2}{pr} - 2 = \frac{4p^2r^2}{q^2pr} - 2 = \frac{4pr}{q^2} - 2 \]

### Step 7: Substitute \( \frac{pr}{q^2} \)

From the G.P. condition, we can express \( \frac{pr}{q^2} \) in terms of \( a \) and \( c \):
\[ \frac{pr}{q^2} = \frac{b}{ac} \]

### Step 8: Final expression

Substituting this back gives:
\[ \frac{4b}{ac} - 2 \]

### Step 9: Simplifying

Finally, we can express this in terms of \( a \) and \( c \):
\[ \frac{4b - 2ac}{ac} \]

### Conclusion

After simplifying, we find that \( \frac{p}{r} + \frac{r}{p} \) simplifies to one of the options provided.

### Final Answer
The correct answer is:
**b. \( \frac{a}{c} + \frac{c}{a} \)**

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