Solve the followings :
If `x^a = y^b = z^c and y^2 = zx`then the value of `1/a + 1/c` is :

Options

A. `b/2`
B. `c/2`
C. `2/b`
D. `2a`

Answer

To solve the equation \( x^a = y^b = z^c \) and \( y^2 = zx \), we can follow these steps:

### Step 1: Express \( x, y, z \) in terms of a common variable
Let \( k = x^a = y^b = z^c \). Then we can express:
- \( x = k^{1/a} \)
- \( y = k^{1/b} \)
- \( z = k^{1/c} \)

### Step 2: Substitute \( x, y, z \) into the second equation
Given the equation \( y^2 = zx \), we substitute the expressions we found:
\[ (k^{1/b})^2 = (k^{1/c})(k^{1/a}) \]
This simplifies to:
\[ k^{2/b} = k^{1/c + 1/a} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ \frac{2}{b} = \frac{1}{c} + \frac{1}{a} \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ \frac{1}{c} + \frac{1}{a} = \frac{2}{b} \]

### Step 5: Find the value of \( \frac{1}{a} + \frac{1}{c} \)
From the rearranged equation, we can directly conclude that:
\[ \frac{1}{a} + \frac{1}{c} = \frac{2}{b} \]

Thus, the value of \( \frac{1}{a} + \frac{1}{c} \) is \( \frac{2}{b} \).

### Final Answer:
\[ \frac{1}{a} + \frac{1}{c} = \frac{2}{b} \]

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