If $A . M$. and $G M$. of two positive numbers $a$ and $b$ are $10$ and $8 , $ respectively, find the numbers.
If $A . M$. and $G M$. of two positive numbers $a$ and $b$ are $10$ and $8 , $ respectively, find the numbers.
Given that $A.M.=$ $=\frac{a+b}{2}=10$ ........$(1)$
and $G.M.=$ $\sqrt{a b}=8$ ........$(2)$
From $(1)$ and $(2),$ we get
$a+b =20$ .........$(3)$
$ a b =64$ .........$(4)$
Putting the value of $a$ and $b$ from $(3),(4)$ in the identity $(a-b)^{2}=(a+b)^{2}-4 a b$
we get $(a-b)^{2}=400-256=144$
or $a-b=\pm 12$ .........$(5)$
Solving $(3)$ and $(5),$ we obtain
$a=4, b=16 \text { or } a=16, b=4$
Thus, the numbers $a$ and $b$ are $4,16$ or $16,4$ respectively.
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