If a,x, and b and b are in A.P .., a,y , and a,z,b are in H.P such that x=9z and `a gt 0, b gt 0` then

Options

A. `y^(2)=xz`
B. `x gt y gt z`
C. `a=9`, `b=1`
D. `a=1//4`, `b=9//4`

Answer

A, B, C

Explanation

`(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`