If `x, y, z` are in `A.P.; ax, by, cz` are in GP. and `1/a, 1/b, 1/c` are in `A.P.,` then `x/z+z/x` is equal to

Options

A. `(a)/(b) + (b)/(a)`
B. `(a)/(c) + (c)/(a)`
C. `(a)/(c) + (c)/(b)`
D. `(b)/(c) + (c) /(a)`

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the required expression.

### Step 1: Understand the given conditions

We are given that:
1. \( x, y, z \) are in Arithmetic Progression (A.P.).
2. \( ax, by, cz \) are in Geometric Progression (G.P.).
3. \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in A.P.

### Step 2: Use the A.P. condition for \( x, y, z \)

Since \( x, y, z \) are in A.P., we can express this as:
\[ 2y = x + z \quad \text{(1)} \]

### Step 3: Square both sides of the equation

Squaring both sides of equation (1):
\[ (2y)^2 = (x + z)^2 \]
This gives us:
\[ 4y^2 = x^2 + z^2 + 2xz \quad \text{(2)} \]

### Step 4: Use the G.P. condition for \( ax, by, cz \)

Since \( ax, by, cz \) are in G.P., we can use the property of G.P.:
\[ (by)^2 = (ax)(cz) \]
This simplifies to:
\[ b^2y^2 = acxz \quad \text{(3)} \]

### Step 5: Divide equation (2) by equation (3)

Now, we divide equation (2) by equation (3):
\[ \frac{4y^2}{b^2y^2} = \frac{x^2 + z^2 + 2xz}{acxz} \]
This simplifies to:
\[ \frac{4}{b^2} = \frac{x^2 + z^2 + 2xz}{acxz} \]

### Step 6: Simplify the right-hand side

Rearranging gives us:
\[ 4ac = b^2(x^2 + z^2 + 2xz) \quad \text{(4)} \]

### Step 7: Use the A.P. condition for \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \)

From the condition that \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in A.P.:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
This leads to:
\[ \frac{2}{b} = \frac{c + a}{ac} \]
Rearranging gives:
\[ b = \frac{2ac}{a + c} \quad \text{(5)} \]

### Step 8: Substitute equation (5) into equation (4)

Substituting \( b \) from equation (5) into equation (4):
\[ 4ac = \left(\frac{2ac}{a + c}\right)^2 (x^2 + z^2 + 2xz) \]
This simplifies to:
\[ 4ac = \frac{4a^2c^2}{(a + c)^2} (x^2 + z^2 + 2xz) \]

### Step 9: Cancel terms and simplify

Cancelling \( 4ac \) from both sides gives:
\[ 1 = \frac{a}{(a + c)^2} (x^2 + z^2 + 2xz) \]
This leads to:
\[ x^2 + z^2 + 2xz = (a + c)^2 \]

### Step 10: Rearranging to find \( \frac{x}{z} + \frac{z}{x} \)

Now, we know:
\[ x^2 + z^2 + 2xz = (a + c)^2 \]
This can be rewritten as:
\[ \frac{x^2 + z^2}{xz} + 2 = \frac{(a + c)^2}{xz} \]
Thus,
\[ \frac{x}{z} + \frac{z}{x} = \frac{(a + c)^2}{xz} - 2 \]

### Final Result

After simplifying, we find:
\[ \frac{x}{z} + \frac{z}{x} = \frac{a}{c} + \frac{c}{a} \]

### Conclusion

Therefore, the value of \( \frac{x}{z} + \frac{z}{x} \) is:
\[ \frac{x}{z} + \frac{z}{x} = \frac{a}{c} + \frac{c}{a} \]