If the G.M. between a and b be twice the H.M., then (a)/(b) is equal to (A) (2+√(3))/(2-√(3)) (B

step 1 Here we have G is equal to under root AB geometric mean, H harmonic mean is equal to 2ab by A pl

If the G.M. between a and b be twice the H.M., then (a)/(b)...

If the G.M. between $a$ and $b$ be twice the H.M., then $\frac{a}{b}$ is equal to
(A) $\frac{2+\sqrt{3}}{2-\sqrt{3}}$
(B) $\frac{2-\sqrt{3}}{2+\sqrt{3}}$
(C) $\frac{4+\sqrt{3}}{4-\sqrt{3}}$
(D) $\frac{4-\sqrt{3}}{4+\sqrt{3}}$

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Key Concepts

Comparison of Means

Comparing different means, such as the geometric and harmonic means, helps reveal intrinsic relationships between the numbers. Establishing a proportional relationship between these means leads to equations that can be solved to determine the ratio of the involved numbers.

Algebraic Manipulation for Ratio Determination

Setting up and solving equations derived from relationships between various means involves algebraic manipulation. This process includes substituting definitions, simplifying equations, and solving for the ratio of the original quantities, which is a common technique in algebra and problem solving.

Geometric Mean

The geometric mean of two positive numbers is defined as the square root of their product. It is often used in situations where growth rates or multiplicative processes are involved, providing a central tendency that is less sensitive to extreme values compared to the arithmetic mean.

Harmonic Mean

The harmonic mean of two positive numbers is calculated as twice the product divided by the sum of the numbers. It is particularly useful in scenarios where rates or ratios need to be averaged, as it tends to give more weight to smaller values than other means.

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