Find the mean proportional for each of the following:
(vi) $2 x, 8 x$

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Solution

Find the mean proportional:
Let $m$ be the mean proportional.
If $m$ is the mean proportional between $a$ and $b$ , then $a : m : : m : b$ .
Then for the numbers $2 x, 8 x$ , the equivalent proportional expression is $2 x : m : : m : 8 x$ .
$\Rightarrow 2 x : m = m : 8 x (C h a n g i n g t h e p r o p o r t i o n a l i t y s i g n t o e q u a l s i g n) \Rightarrow \frac{2 x}{m} = \frac{m}{8 x} \Rightarrow m \cdot m = 2 x \cdot 8 x (C r o s s m u l t i p l y i n g b o t h s i d e s) \Rightarrow m^{2} = 16 x^{2} ∴ m = \pm 4 x (T a k i n g s q u a r e r o o t o n b o t h s i d e s)$
As $- 4 x$ does not lie between the given two numbers, therefore, the mean proportional is $4 x$ .
Final Answer: The mean proportional is $4 x$ .

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