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If a,x,b are in A.P.,a,y,b are in G.P. a...

If a,x,b are in A.P.,a,y,b are in G.P. and a,z,b are in H.P. such that x=9z and a>0, b>0, then (a) y=3z. (b) x=3y (c) 2y= x+z (d) none of these

A

`y^(2)=xz`

B

`x gt y gt z`

C

`a=9`, `b=1`

D

`a=1//4`, `b=9//4`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(a,b,c)` `2x=a+b`
`y^(2)=ab`
`z=(2ab)/(a+b)`
`x=y+2`
and `a=5z`
`z=(2y^(2))/(2x)`
`impliesy^(2)=xz`
`x=y+2`
`:.(a+b)/(2)=sqrt(ab)+2`………`(i)`
and `a=5((2ab))/(a+b)`
`implies(a+b)=10b`
`impliesa=9b`
`:.` from `(i) (9b+b)/(2)=sqrt(9b^(2))+2`
`implies5b=3b+2`
`impliesb=1`
So `a=9`
`implies x gt y gt z`
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