If `a, b, c` are in `G.P`. and `x, y` are arithmetic means of `a, b` and `b, c` respectively, then `(1)/(x)+(1)/(y)` is equal to

Options

A. `(2)/(b)`
B. `(3)/(b)`
C. `(b)/(3)`
D. `(b)/(2)`

Answer

To solve the problem step by step, we start with the given conditions and proceed to find the value of \( \frac{1}{x} + \frac{1}{y} \).

### Step 1: Understand the Given Conditions
We know that \( a, b, c \) are in geometric progression (G.P.). This means:
\[ b^2 = ac \tag{1} \]

### Step 2: Find the Arithmetic Means
Next, we define \( x \) and \( y \) as the arithmetic means:
- \( x \) is the arithmetic mean of \( a \) and \( b \):
\[ x = \frac{a + b}{2} \tag{2} \]
- \( y \) is the arithmetic mean of \( b \) and \( c \):
\[ y = \frac{b + c}{2} \tag{3} \]

### Step 3: Calculate \( \frac{1}{x} + \frac{1}{y} \)
We want to find \( \frac{1}{x} + \frac{1}{y} \):
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} \]

### Step 4: Substitute Values of \( x \) and \( y \)
Substituting the values of \( x \) and \( y \) from equations (2) and (3):
\[ \frac{1}{x} + \frac{1}{y} = \frac{\frac{b + c}{2} + \frac{a + b}{2}}{\left(\frac{a + b}{2}\right) \left(\frac{b + c}{2}\right)} \]
This simplifies to:
\[ = \frac{\frac{(b + c) + (a + b)}{2}}{\frac{(a + b)(b + c)}{4}} = \frac{(a + 2b + c)}{(a + b)(b + c)} \cdot 2 \]

### Step 5: Simplify the Expression
Now we have:
\[ \frac{1}{x} + \frac{1}{y} = \frac{2(a + 2b + c)}{(a + b)(b + c)} \]

### Step 6: Substitute \( b^2 = ac \)
Using equation (1), we can substitute \( ac \) with \( b^2 \):
- The denominator becomes:
\[ (a + b)(b + c) = ab + ac + b^2 + bc = ab + b^2 + b^2 + bc = ab + 2b^2 + bc \]

### Step 7: Final Simplification
Now, we can rewrite the expression:
\[ \frac{2(a + 2b + c)}{ab + 2b^2 + bc} \]
Since \( b^2 = ac \), we can further simplify:
\[ = \frac{2(a + 2b + c)}{ab + ac + bc} = \frac{2(a + 2b + c)}{b(a + c) + ac} \]

### Step 8: Conclude the Result
After simplification, we find that:
\[ \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \]

Thus, the final answer is:
\[ \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \]

---