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

Options

A. `|y|=3zandx=3|y|`
B. `y=3|z|and|x|=3y`
C. 2y=x+z
D. none of these

Answer

A

Explanation

It is given that
a,x,b are in A.P. `rArr` x is the A.M. of a and b.
a,y,b are in G.P. `rArr` y is the G.M. of and b.
a,z,b are in H.P `rArr` z is the H.M. of a and b.
Thus, x,y and are respectively the A.M., G.M. and H.M. of a and b. Also, A.M., G.M. and H.M. are in G.P.
`:." "y^(2)=xz`
`rArr" "y^(2)=9z^(2)" "[becausex=9z]`
`rArr" "y=3zrArr|y|=3z" "[becauseygt0:.|y|=y]`
Again,
`rArr" "y^(2)=xzandx=9z`
`rArr" "y^(2)=(x^(2))/(9)`
`rArr" "9y^(2)=x^(2)rArr3y=xrArr3|y|=x" "[becauseygt0:.|y|=y]`