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The sum of two numbers is 6 times their ...

The sum of two numbers is 6 times their geometric means, show that numbers are in the ratio `(3+2sqrt(2)):(3-2sqrt(2))`.

A

`6:1`

B

`3:2`

C

`2:1`

D

`3:1`

Text Solution

Verified by Experts

The correct Answer is:
d
Let two positive be a and b
` :. a/b = (3+2sqrt(2))/(3-2sqrt(2)) xx(3+2sqrt(2))/(3+2sqrt(2) )= (9+8+12sqrt(2))/(9-8) = 17 + 12 sqrt(2)`
Now, ` (AM)/(GM) = ((a+b)/2)/(sqrt(ab))= ((17 +12 sqrt(2))b+b)/(2sqrt((17+12sqrt(2))b))=(18+12sqrt(2))/(2sqrt(17+12sqrt(2)))`
` = (9+6sqrt(2))/(sqrt((3+2sqrt(2))^(2)))=(3(3+2sqrt(2)))/(3+2sqrt(2))=3/1`
` :. AM :GM = 3:1`
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