If `tan^-1x, tan^-1y` and `tan^-1z` are in A.P. then find the algebraic relation between x,y and z. If `x,y,z` are also in A.P. then show that `x=y=z` and `y!=0`

Options

A. `x=y=z`
B. `x=y=-z`
C. `x=1,y=2,z=3`
D. `x=2,y=4,z=6`

Answer

A

Explanation

Since x,y,z are in AP
`y=(x+z)/2`…….i
and `tan^(-1)x, tan^(-1)y` and `tan^(-1)z` are also in AP.
`:.2tan^(-1)y=tan^(-1)x+tan^(-1)z`
`impliestan^(-1)((2y)/(1-y^(2)))="tan"^(-1)(x+z)/(1-xz)`
`implies(2y)/(1-y^(2))=(2h)/(1-xz)` [from eq. (i) ]
`impliesy^(2)=xzimpliesx,y,z` are in GP
`:.x=y=z`