Question

If a is positive and also \[A,G\] are the arithmetic mean and the geometric mean of the roots of \[{{x}^{2}}-2ax+{{a}^{2}}=0\] respectively, then
\[\begin{align}
  & \left( 1 \right)A=G \\
 & \left( 2 \right)A=2G \\
 & \left( 3 \right)2A=G \\
 & \left( 4 \right){{A}^{2}}=G \\
\end{align}\]

Answer 1

Hint: In order to solve the given problem, we must be considering the roots of the given quadratic equation. After considering them, we will be applying the formulas of both arithmetic mean as well as geometric mean to the roots of the given equation. Then we cross check with the answer obtained with the given options and conclude the answer.

Complete step-by-step solution:
Now let us learn more about the arithmetic mean and geometric mean. Arithmetic mean is nothing but the value obtained by adding up the observations and dividing it with the number of observations. Geometric mean is nothing but for a given number of values containing \[n\] observations is the \[{{n}^{th}}\] root for the product of the values. There exist a relation between the arithmetic mean, harmonic mean and geometric mean and i.e. \[AM\times HM=G{{M}^{2}}\].
Now let us start solving our given problem.
Firstly, we will be finding the roots of the given equation \[{{x}^{2}}-2ax+{{a}^{2}}=0\]
By looking at the formula, we can simplify it in the form of an identity and i.e. \[{{\left( x-a \right)}^{2}}=0\]
Now let us solve it in order to find the roots of the equation.
\[\begin{align}
  & {{\left( x-a \right)}^{2}}=0 \\
 & \Rightarrow x-a=0 \\
 & \Rightarrow x=a \\
\end{align}\]
We get both the roots of the given equation as \[a\].
We know that the formula of arithmetic mean is \[\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}...{{a}_{n}}}{n}=AM\] and also the formula of geometric mean is \[n\sqrt{{{a}_{1}}{{a}_{2}}{{a}_{3}}...{{a}_{n}}}=GM\].
Now let us apply these formulas for the roots of the equation.
We get,
\[AM=\dfrac{a+a}{2}=a\] and
\[GM=\sqrt{a\times a}=a\]
We can see that for the given equation both arithmetic mean and geometric mean are equal.
Hence option 1 is the correct answer.

Note: We must always try to simplify the given equation in order to find the roots or else we can express it in the form of identity if possible as we have done in the problem. The formulas of the AM and GM are supposed to be applied correctly. If we are also provided with HM, we must be applying the general relation between the AM, GM and HM.