If `a ,b , a n dc` are in G.P. and `x ,y ,` respectively, are the arithmetic means between `a ,b ,a n db ,c` , then the value of `a/x+c/y` is `1` b. `2` c. `1//2` d. none of these

Options

A. 1
B. 2
C. `1//2`
D. none of these

Explanation

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

### Step 1: Understanding the Given Information
We know that \( a, b, c \) are in Geometric Progression (G.P.). This means that:
\[ \frac{b}{a} = \frac{c}{b} \]
From this, we can express \( b \) and \( c \) in terms of \( a \):
\[ b^2 = ac \quad \text{(1)} \]

### Step 2: Finding the Arithmetic Means
We are given that \( x \) and \( y \) are the arithmetic means between \( a, b \) and \( b, c \) respectively. Thus:
\[ x = \frac{a + b}{2} \quad \text{(2)} \]
\[ y = \frac{b + c}{2} \quad \text{(3)} \]

### Step 3: Finding \( \frac{a}{x} + \frac{c}{y} \)
We need to calculate \( \frac{a}{x} + \frac{c}{y} \). Using equations (2) and (3), we substitute \( x \) and \( y \):
\[ \frac{a}{x} = \frac{a}{\frac{a + b}{2}} = \frac{2a}{a + b} \]
\[ \frac{c}{y} = \frac{c}{\frac{b + c}{2}} = \frac{2c}{b + c} \]

Now, we can combine these two fractions:
\[ \frac{a}{x} + \frac{c}{y} = \frac{2a}{a + b} + \frac{2c}{b + c} \]

### Step 4: Finding a Common Denominator
The common denominator for the fractions is \( (a + b)(b + c) \). Thus, we rewrite the expression:
\[ \frac{2a(b + c) + 2c(a + b)}{(a + b)(b + c)} \]

### Step 5: Simplifying the Numerator
Expanding the numerator:
\[ 2a(b + c) + 2c(a + b) = 2ab + 2ac + 2ca + 2cb = 2ab + 2ac + 2bc \]

### Step 6: Using the G.P. Condition
From the G.P. condition \( b^2 = ac \), we can substitute \( ac \) with \( b^2 \):
\[ = 2ab + 2b^2 + 2bc \]

### Step 7: Simplifying the Denominator
The denominator is:
\[ (a + b)(b + c) = ab + ac + b^2 + bc \]

### Step 8: Final Expression
Now we have:
\[ \frac{2(ab + b^2 + bc)}{ab + ac + b^2 + bc} \]

### Step 9: Evaluating the Expression
Notice that if we factor out the common terms:
\[ = 2 \cdot \frac{ab + b^2 + bc}{ab + ac + b^2 + bc} \]

Since \( b^2 = ac \), the expression simplifies to:
\[ = 2 \cdot \frac{ab + ac}{ab + ac + b^2} \]

This leads us to conclude:
\[ = 2 \]

### Conclusion
Thus, the value of \( \frac{a}{x} + \frac{c}{y} \) is \( 2 \).

### Final Answer
The answer is \( \text{b. } 2 \).

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