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If AM and GM of x and y are in the ratio...

If AM and GM of x and y are in the ratio of p:q, then x:y is

A

` p - sqrt(p^(2)+q^(2)):sqrt(p^(2)+q^(2))`

B

` p+sqrt(p^(2)-q^(2)) : p - sqrt(p^(2)-q^(2))`

C

`p :q`

D

` p+ sqrt(p^(2)+q^(2)) : p - sqrt(p^(2)+q^(2))`

Text Solution

Verified by Experts

The correct Answer is:
b
Since ` (x+y)/(2sqrt(xy) ) = p/q`
` rArr ((x+y)^(2))/(4xy) = (p^(2))/(q^(2))" " ` …(i)
On subtracting both sides by 1 , we get
` (x-y)^(2)/(4xy) = (p^(2)-q^(2))/(q^(2))" "` … (ii)
From Eqs. (i) and (ii) ,
` (x+y)/(x-y)^(2)= (p^(2))/(p^(2)-q^(2))`
`rArr (x+y)/(x-y) =p/(sqrt(p^(2)-q^(2)))`
` rArr (2x)/(2y) = (p+sqrt(p^(2)-q^(2)))/(p-sqrt(p^(2)-q^(2)))`
[ by componendo - dividendo rule ]
` :. x : y = (p+sqrt(p^(2)-q^(2))): (p - sqrt(p^(2)-q^(2)))`
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