Question

The A.M between m and n and the G.M between a and b are each equal to (ma+nb)/(m+n). Find m in terms of a and b.
$
  (a){\text{ }}\dfrac{{\sqrt {ab} }}{{\sqrt a + \sqrt b }}.\sqrt a \\
  (b){\text{ }}\dfrac{{\sqrt {ab} }}{{\sqrt a + \sqrt b }}.\sqrt b \\
  (c){\text{ }}\dfrac{{2\sqrt {ab} }}{{\sqrt a + \sqrt b }}.\sqrt b \\
  (d){\text{ }}\dfrac{{2\sqrt {ab} }}{{\sqrt a + \sqrt b }}.\sqrt a \\
$

Answer 1

Hint – In this question use the concept that A.M of two numbers m and n is $\left( {\dfrac{{m + n}}{2}} \right)$ and the G.M between two numbers m and n is$\sqrt {mn} $. Use this to formulate the relation between m, a and b.

Complete step-by-step answer:
As we know that arithmetic mean (A.M) of m and n is =$\left( {\dfrac{{m + n}}{2}} \right)$..................... (1)
And we also know that geometric mean (G.M) of a and b is =$\sqrt {ab} $.................... (2)
Now according to question
$A.M = \dfrac{{\left( {ma + nb} \right)}}{{m + n}}$
And $G.M = \dfrac{{\left( {ma + nb} \right)}}{{m + n}}$
Now from equation (1) and (2) we have,
$ \Rightarrow \dfrac{{m + n}}{2} = \dfrac{{\left( {ma + nb} \right)}}{{m + n}}$................... (3)
And
$ \Rightarrow \sqrt {ab} = \dfrac{{\left( {ma + nb} \right)}}{{m + n}}$........................... (4)
$ \Rightarrow \dfrac{{m + n}}{2} = \sqrt {ab} $
Now calcite the value of n
$ \Rightarrow m + n = 2\sqrt {ab} $
$ \Rightarrow n = 2\sqrt {ab} - m$
Now substitute this value in equation (3) we have,
$ \Rightarrow \dfrac{{m + 2\sqrt {ab} - m}}{2} = \dfrac{{\left( {ma + b\left( {2\sqrt {ab} - m} \right)} \right)}}{{m + 2\sqrt {ab} - m}}$
Now simplify the above equation we have,
$ \Rightarrow \dfrac{{\sqrt {ab} }}{1} = \dfrac{{\left( {ma + b\left( {2\sqrt {ab} - m} \right)} \right)}}{{2\sqrt {ab} }}$
$ \Rightarrow 2ab = ma + 2b\sqrt {ab} - mb$
$ \Rightarrow m\left( {a - b} \right) = 2ab - 2b\sqrt {ab} $
$ \Rightarrow m = \dfrac{{2\sqrt {ab} \left( {\sqrt {ab} - b} \right)}}{{a - b}}$
Now again take $\sqrt b $ common from numerator and (a – b) is written as $\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)$
$ \Rightarrow m = \dfrac{{2\sqrt {ab} \sqrt b \left( {\sqrt a - \sqrt b } \right)}}{{\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)}}$
$ \Rightarrow m = \dfrac{{2\sqrt {ab} }}{{\sqrt a + \sqrt b }}.\sqrt b $
So this is the required value of m in terms of a and b.
Hence option (C) is correct.

Note – Arithmetic mean is the average of a set of numerical values, as calculated by adding them together and dividing by the number of terms in the set, they arithmetic mean of n numbers ${a_1},{a_2}..............{a_n}$ is $\dfrac{{{a_1} + {a_2} + {a_3} + .......... + {a_n}}}{n}$, Geometric mean is the central number in a geometric progression, and also calculated as the nth root of product of n numbers, that is geometric mean of ${a_1},{a_2}..............{a_n}$ is $^n\sqrt {{a_1}{a_2}.............{a_n}} $.