Question

If \[{\rm{A}}\] and \[{\rm{G}}\] are arithmetic and geometric means and \[{x^2} - 2Ax + {G^2} = 0\] then
A. \[A = G\]
B. \[A > G\]
C. \[A < G\]
D. \[A = - G\]

Answer 1

Hint:
We must examine the roots of the provided quadratic equation in order to answer the given problem. After considering these, we will apply the arithmetic and geometric mean formulas to the roots of the given equation. Then we have to cross-check the response with the offered possibilities and come to a conclusion.
Formula used:
\[AM = \frac{{{a_1} + {a_2} + {a_3}...{a_n}}}{n}\]
\[GM = \sqrt[n]{{{a_1} + {a_2} + {a_3}...{a_n}}}\]
Complete Step-by-Step Solution:
Let us now understand more about the arithmetic and geometric means. Arithmetic mean is the result produced by summing the data and dividing by the number of observations. Geometric mean is nothing more than the nth element of the product of the values for a certain number of values including data instances.
We have been provided in the question that,
\[{\rm{A}}\] And \[{\rm{G}}\] are arithmetic and geometric means and
\[{x^2} - 2Ax + {G^2} = 0\]
Let us assume that \[\alpha \] and \[\beta \] are the roots of the equation
\[{x^2} - 2ax + {a^2} = 0\]
As below mentioned, the equation form the previous calculation can be written as,
\[\alpha + \beta = 2a{\rm{ and }}\alpha \beta = {a^2}\]----- (1)
As we know that, A and G are the arithmetic and geometric mean of the roots.
That is,
\[A = \frac{{\alpha + \beta }}{2}\]
Or
\[G = \sqrt {\alpha \beta } \]
Now, from the equation (1), we obtain the expression as below
\[\frac{{\alpha + \beta }}{2} = a\]
And
\[\alpha \beta = {a^2}\]
The previously obtained equations can be written as following,
\[ \Rightarrow A = a\]
And
\[{G^2} = {a^2}\]
On simplifying the above expressions, we get
\[ \Rightarrow {G^2} = {A^2} \Rightarrow G = A\]
Therefore, if \[{\rm{A}}\] and \[{\rm{G}}\] are arithmetic and geometric means and \[{x^2} - 2Ax + {G^2} = 0\] then \[A = G\]
Hence, the option A is correct
Note:
We should always strive to simplify the provided equation in order to locate the roots, or else represent it as an identity if possible, like we did in the issue. The AM and GM formulas are expected to be used correctly. If we are also given HM, we must use the general relationship between the AM, GM, and HM.