Let x, y, z be positive real numbers such that x, y, z are in GP and `tan^(-1)y" and "tan^(-1)z` are in AP. Then which one of the following is correct?

Options

A. `x=y=z`
B. `xz=1`
C. `xney" and "y=z`
D. `x=y" and "ynez`

Answer

A

Explanation

`2tan^(-1)y=tan^(-1)x+tan^(-1)z`
`( :. tan^(-1)x,tan^(-1)y,tan^(-1)z" are in A.P")`
and x,y are in G.P`" ":.y^(2)=xz" "...(i)`
`implies(2y)/(1-y^(2))=(x+z)/(1-xz)`
`implies(2y)/(cancel(1-y^(2)))=(x+y)/(cancel(1-y^(2)))" "("from (i)")`
`implies2y=x+z`
x,y,z are in A.P.
Given x, y, z are also in G.P
So, x=y=z