Question

If A and G be A.M andG.M.,respectively between two positive number, prive that the numbers are
$A+-\sqrt{(A+G)}(A+G)$

Answer 1

Let, the nos. be $x\&y$
A.M of $x\&y=\frac{x+y}{2}=A$ (given) --------(1)
G.M. of $x\&y=\sqrt{xy}=G$ (given) ------(2)
Now, you how to write x & y in terms of A & G
From (1) x=2A -y
Substituting in (2) $xy=g^2\Rightarrow (2A-y)y=G^2$
$\Rightarrow -y^2+2Ay-G^2=0$
$\Rightarrow y^2-2Ay+G^2=0$
$y=\frac{2A\pm \sqrt{4A^2-4G^2}}{2}=A+\sqrt{A^2-G^2}\&A-\sqrt{A^2-G^2}$
$\therefore x=A-\sqrt{A^2-G^2}orA+\sqrt{A^2-G^2}$
So, the nos. are
$A\pm \sqrt{(A+G)(A+G)}$
proved
$y=\frac{2A\pm \sqrt{4A^2-4G^2}}{2}=A+\sqrt{A^2-G^2}\&A+\sqrt{A^2-G^2}\therefore x=A-\sqrt{A^2-G^2}orA+\sqrt{A^2-G^2}$