$ ext{If } a^x = b^y = c^z ext{ and } b^2 = ac ext{, then the value of } y ext{ is :}$

Q: $\text{If } a^x = b^y = c^z \text{ and } b^2 = ac \text{, then the value of } y \text{ is :}$

To solve the problem, we need to find the value of $ y $ given the conditions $ a^x = b^y = c^z $ and $ b^2 = ac $. ### Step-by-step Explanation: 1. **Equating the Powers:** Since $ a^x = b^y = c^z = k $ (say), we can express each variable in terms of $ k $: \[ a = k^{1/x}, \quad b = k^{1/y}, \quad c = k^{1/z} \] 2. **Using the Condition $ b^2 = ac $:** Substitute the expressions for $ a $, $ b $, and $ c $ into the equation: \[ (k^{1/y})^2 = k^{1/x} \cdot k^{1/z} \] Simplifying the left side: \[ k^{2/y} = k^{1/x + 1/z} \] 3. **Equating the Exponents:** Since the bases are the same, equate the exponents: \[ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} \] 4. **Solving for $ y $:** To find $ y $, take the reciprocal of both sides: \[ y = \frac{2}{\frac{1}{x} + \frac{1}{z}} \] Simplifying the right side: \[ y = \frac{2}{\frac{x + z}{xz}} = \frac{2xz}{x + z} \] Thus, the value of $ y $ is $\frac{2xz}{x + z}$, which corresponds to **Option 1**. ### Explanation of Incorrect Options: - **Option 2: $\frac{xz}{x + z}$** - This option does not account for the factor of 2 in the numerator, which arises from the equation $ b^2 = ac $. The correct derivation shows that the factor of 2 is necessary to balance the equation. - **Option 3: $\frac{xz}{2x + z}$** - This option incorrectly modifies the denominator. The correct denominator should be $ x + z $, as derived from the equation $ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} $. - **Option 4: $\frac{3xz}{x + z}$** - This option introduces an incorrect factor of 3 in the numerator. The derivation clearly shows that the factor should be 2, based on the given condition $ b^2 = ac $. In conclusion, the correct answer is indeed **Option 1: $\frac{2xz}{x + z}$**, as it accurately reflects the relationship derived from the given conditions.

Question

$	ext{If } a^x = b^y = c^z 	ext{ and } b^2 = ac 	ext{, then the value of } y 	ext{ is :}$

Accepted Answer

To solve the problem, we need to find the value of $ y $ given the conditions $ a^x = b^y = c^z $ and $ b^2 = ac $.

### Step-by-step Explanation:

1. **Equating the Powers:**
   Since $ a^x = b^y = c^z = k $ (say), we can express each variable in terms of $ k $:
   $$
   a = k^{1/x}, \quad b = k^{1/y}, \quad c = k^{1/z}
   $$

2. **Using the Condition $ b^2 = ac $:**
   Substitute the expressions for $ a $, $ b $, and $ c $ into the equation:
   $$
   (k^{1/y})^2 = k^{1/x} \cdot k^{1/z}
   $$
   Simplifying the left side:
   $$
   k^{2/y} = k^{1/x + 1/z}
   $$

3. **Equating the Exponents:**
   Since the bases are the same, equate the exponents:
   $$
   \frac{2}{y} = \frac{1}{x} + \frac{1}{z}
   $$

4. **Solving for $ y $:**
   To find $ y $, take the reciprocal of both sides:
   $$
   y = \frac{2}{\frac{1}{x} + \frac{1}{z}}
   $$
   Simplifying the right side:
   $$
   y = \frac{2}{\frac{x + z}{xz}} = \frac{2xz}{x + z}
   $$

Thus, the value of $ y $ is $\frac{2xz}{x + z}$, which corresponds to **Option 1**.

### Explanation of Incorrect Options:

- **Option 2: $\frac{xz}{x + z}$**
  - This option does not account for the factor of 2 in the numerator, which arises from the equation $ b^2 = ac $. The correct derivation shows that the factor of 2 is necessary to balance the equation.

- **Option 3: $\frac{xz}{2x + z}$**
  - This option incorrectly modifies the denominator. The correct denominator should be $ x + z $, as derived from the equation $ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} $.

- **Option 4: $\frac{3xz}{x + z}$**
  - This option introduces an incorrect factor of 3 in the numerator. The derivation clearly shows that the factor should be 2, based on the given condition $ b^2 = ac $.

In conclusion, the correct answer is indeed **Option 1: $\frac{2xz}{x + z}$**, as it accurately reflects the relationship derived from the given conditions.