if `x_(1) gt 0,y_(1) gt 0 (1,2 3…. n)` are the values of two variable X and Y with geometric mean P and Q respectively, then the geometric mean of `(x)/(y)` is

Options

A. `(P)/(Q)`
B. antilog `(P)/(Q)`
C. a(log P-log Q)
D. a (log P+log Q)

Answer

To solve the problem step by step, we need to find the geometric mean of the ratio \( \frac{x}{y} \) given that the geometric means of \( x \) and \( y \) are \( P \) and \( Q \) respectively.

### Step 1: Understand the Geometric Mean
The geometric mean of a set of values \( x_1, x_2, \ldots, x_n \) is defined as:
\[ P = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} \]
Similarly, for the values \( y_1, y_2, \ldots, y_n \):
\[ Q = \sqrt[n]{y_1 \cdot y_2 \cdot \ldots \cdot y_n} \]

### Step 2: Write the Geometric Mean of the Ratio
We need to find the geometric mean of the ratio \( \frac{x}{y} \). The geometric mean of the ratios can be expressed as:
\[ \text{Geometric Mean of } \frac{x}{y} = \sqrt[n]{\frac{x_1}{y_1} \cdot \frac{x_2}{y_2} \cdots \frac{x_n}{y_n}} \]

### Step 3: Rewrite the Expression
This can be rewritten using the properties of exponents:
\[ \text{Geometric Mean of } \frac{x}{y} = \sqrt[n]{\frac{x_1 \cdot x_2 \cdots x_n}{y_1 \cdot y_2 \cdots y_n}} = \frac{\sqrt[n]{x_1 \cdot x_2 \cdots x_n}}{\sqrt[n]{y_1 \cdot y_2 \cdots y_n}} \]

### Step 4: Substitute the Geometric Means
From our definitions of \( P \) and \( Q \):
\[ \text{Geometric Mean of } \frac{x}{y} = \frac{P}{Q} \]

### Conclusion
Thus, the geometric mean of \( \frac{x}{y} \) is:
\[ \frac{P}{Q} \]

### Final Answer
The geometric mean of \( \frac{x}{y} \) is \( \frac{P}{Q} \).

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