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If `x, y, z` are distinct positive real numbers is A.P. then `(1)/(sqrt(x)+sqrt(y)), (1)/(sqrt(z)+sqrt(x)), (1)/(sqrt(y)+sqrt(z))` are in

A

`A.P.`

B

`G.P.`

C

`H.P.`

D

`A.G.P.`

Text Solution

Verified by Experts

The correct Answer is:
A
If `x, y, z` are ………….
`x, y, z` are in `AP , y - x = z - y`
`(sqrt(y)+sqrt(x))(sqrt(y)-sqrt(x))=(sqrt(z)+sqrt(y))`
`(sqrt(z)-sqrt(y))`
`(sqrt(y)-sqrt(x))/(sqrt(y)+sqrt(z)) = (sqrt(z)-sqrt(y))/(sqrt(y)+sqrt(x))`
`((sqrt(y)+sqrt(z))-(sqrt(z)+sqrt(x)))/(sqrt(y)+sqrt(z))`
`= ((sqrt(z)+sqrt(x))-(sqrt(x)+sqrt(y)))/((sqrt(x)+sqrt(y)))`
`1- (sqrt(z)+sqrt(x))/(sqrt(y)+sqrt(z)) = (sqrt(z)+sqrt(x))/(sqrt(x)+sqrt(y)) - 1`
`2 = (sqrt(z)+sqrt(x)) ((1)/(sqrt(x)+sqrt(y))+(1)/(sqrt(y)+sqrt(z)))`
`(2)/(sqrt(z)+sqrt(x)) = (1)/(sqrt(x)+sqrt(y)) + (1)/(sqrt(y)+sqrt(z))`
`:.` Number are in `A.P`
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