If two vriables a and b are related by c= ab then GM.of c =(a) GM of a +GM of b(b) GM of a×GM of b(c) GM of a -GM of b(d) GM of a /GM of b

To determine how the geometric mean (GM) of $c$ relates to the geometric means of $a$ and $b$ given the relationship $c = ab$, let’s break down the problem and analyze the options.

1. Break Down the Problem

• We have $c = ab$.
• We want to find out how $\text{GM}(c)$ relates to $\text{GM}(a)$ and $\text{GM}(b)$.

2. Relevant Concepts

• The geometric mean of a set of numbers $x_1, x_2, \ldots, x_n$ is given by: $\text{GM}(x_1, x_2, \ldots, x_n) = (x_1 x_2 \cdots x_n)^{1/n}$

• Specifically, for two variables $a$ and $b$, the geometric mean is: $\text{GM}(a, b) = (ab)^{1/2}$

3. Analysis and Detail

• Using the definition, we calculate: $\text{GM}(c) = \text{GM}(ab) = (ab)^{1/2}$

• Now, calculate $\text{GM}(a)$, $\text{GM}(b)$: $\text{GM}(a) = (a)^{1/1} = a$ $\text{GM}(b) = (b)^{1/1} = b$

4. Relation between GM of c, a, and b

• Now we need to express this in terms of $\text{GM}(a)$ and $\text{GM}(b)$: $\text{GM}(c) = \sqrt{ab} = \sqrt{\text{GM}(a) \cdot \text{GM}(b)}$

This suggests that $\text{GM}(c)$ is not straightforwardly expressed by any of the linear combinations indicated in the options.

Final Answer

The formulation relates to $\text{GM}(a)$ and $\text{GM}(b)$ as: $\text{GM}(c) = \sqrt{(a)(b)} = \sqrt{\text{GM}(a) \times \text{GM}(b)}$ Thus, the correct option is:

• (b) GM of a × GM of b

Therefore, the correct answer is $(b) \text{GM}(a) \times \text{GM}(b)$.