Question

If the A.M. is twice the G.M. of the numbers \[a\] and \[b\], then \[a:b\] will be
A. \[\frac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }}\]
B. \[\frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\]
C. \[\frac{{\sqrt 3 - 2}}{{\sqrt 3 + 2}}\]
D. \[\frac{{\sqrt 3 + 2}}{{\sqrt 3 - 2}}\]

Answer 1

Hint:
We should first define AM and GM before we can answer this query. When two numbers are a and b, \[{\rm{AM}} = \frac{{a + b}}{2}\]and \[{\rm{GM}} = \sqrt {ab} \]. The geometric mean, also known as the geometric mean (GM), is the sum of all terms in a geometric sequence, while the arithmetic mean is the average of two numbers.
Formula used:
\[{\rm{AM}} = \frac{{a + b}}{2}\]and \[{\rm{GM}} = \sqrt {ab} \]
Complete step-by-step solution:
The given condition is that A.M is double the G.M
\[{\rm{A}}{\rm{. }}M = 2G \cdot M\]
Now replace the formula for A.M and G.M
\[ \Rightarrow \frac{{a + b}}{2} = 2\sqrt {ab} \]
Now, we have to move all the variables to one side and having constant on other side, we get
\[ \Rightarrow \frac{{a + b}}{{2\sqrt {ab} }} = 2\]
Now, we have to conjugate the above expression, we get
\[ \Rightarrow \frac{{a + b + 2\sqrt {ab} }}{{a + b - 2\sqrt {ab} }} = \frac{{2 + 1}}{{2 - 1}}\]
Now, simplify using square formula:
\[ \Rightarrow \frac{{{{(\sqrt a + \sqrt b )}^2}}}{{{{(\sqrt a - \sqrt b )}^2}}} = \frac{3}{1}\]
Rewrite the above expression, we obtain
\[ \Rightarrow \frac{{(\sqrt a + \sqrt b )}}{{(\sqrt a - \sqrt b )}} = \frac{{\sqrt 3 }}{1}\]
Now, we have to solve for\[\frac{{\sqrt a }}{{\sqrt b }}\]:
\[\frac{{\sqrt a }}{{\sqrt b }} = \frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}\]
Now, take square on both sides, we get
\[\frac{a}{b} = \frac{{{{(\sqrt 3 + 1)}^2}}}{{{{(\sqrt 3 - 1)}^2}}}\frac{1}{1}{\rm{ }}\]
Solve the square using square formula, we get
\[ = \frac{{3 + 2\sqrt 3 + 1}}{{3 - 2\sqrt 3 + 1}}\frac{1}{1}\]
Simplify the values on the numerator and denominator:
\[ = \frac{{4 + 2\sqrt 3 }}{{4 - 2\sqrt 3 }}\frac{1}{1}\]
Now, we have to simplify the above equation, we get
\[ = \frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\]
Therefore, if the A.M. is twice the G.M. of the numbers\[a\]and\[b\], then\[a:b\]will be\[\frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\]
Hence, the option B is correct.
Note:
The point in this question where the student must solve the problem without taking conjugate, where there is a chance for error and also while equating the two formulas. Student must be careful while solving Harmonic progression and Geometric progression problems.