the sum of two numbers is six times their geometric mean show that the number are in the ratio a:b and also find a and b
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Step-by-step explanation:
Let the two numbers are a and b.
Thus their G.M=
ab
According to the given condition, a+b=6
ab
....(1)
⇒(a+b)
2
=36(ab)
Also, (a−b)
2
=(a+b)
2
−4ab=36ab−4ab=32ab
⇒a−b=
32
ab
=4
2
ab
........(2)
Adding (1) and (2) , we obtain 2a=(6+4
2
)
ab
⇒a=(3+2
2
)
ab
Substituting the value of a in (1), we obtain b=6
ab
−(3+2
2
)
ab
⇒b=(3−2
2
)
ab
⇒
b
a
=
(3−2
2
)
ab
(3+2
2)
ab
=
3−2
2
3+2
2
Thus , the required ratio is (3+2
2
):(3−2
2
)
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