Question

If the Arithmetic Mean (AM) of two numbers is $A$ and Geomatics Mean (GM) is $G$, then the numbers will be

• A. $A\pm (A^2-G^2)$
• B. $\sqrt{A}\pm \sqrt{(A^2-G^2)}$
• C. $A\pm \sqrt{(A+G)(A-G)}$
• D. $\frac{(A\pm \sqrt{(A+G)(A-G)})}{2}$

Answer 1

The correct option is C

$A\pm \sqrt{(A+G)(A-G)}$

Explanation for Correct Option

Step 1: Apply the arithmetic mean and geometric mean formula

Let $a,b$be the numbers

$$\begin{gathered} \therefore A=\frac{\left(\mathrm{a}+\mathrm{b}\right)}{2} \\ \Rightarrow 2A=a+b.....\left(\mathrm{i}\right) \end{gathered}$$

$$\begin{gathered} \therefore G=\sqrt{\mathrm{ab}} \\ \Rightarrow G^2=ab....\left(\mathrm{ii}\right) \end{gathered}$$

Step 2: Use the Quadratic formula

Quadratic equation with $a,b$ as root is $x^2-(a+b)x+ab=0$

Compare quadratic equation with equation (i) and (ii) we get,

$(a+b)=2A$ and $ab=G^2$

Put the above value in a quadratic equation

$\Rightarrow x^2-2Ax+G^2=0$

$$\begin{gathered} \Rightarrow x=\frac{2A\pm \sqrt{4A^2-G^2}}{2} \\ \Rightarrow x=\frac{2A\pm 2\sqrt{A^2-G^2}}{2} \\ \Rightarrow x=A\pm \sqrt{A^2-G^2} \end{gathered}$$

So, numbers will be $x=A\pm \sqrt{(A-G)(A+G)}$

Hence option (3) is the Correct option