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If x, y, z are in A.P. and t a n^(-1)x...

If x, y, z are in A.P. and `t a n^(-1)x ,""t a n^(-1)y"" and ""t a n^(-1)z` are also in A.P., then (1) `2x""=""3y""=""6z` (2) `6x""=""3y""=""2z` (3) `6x""=""4y""=""3z` (4) `x""=""y""=""z`

Text Solution

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`2y = x + z`
`2 tan^-1 y = tan^-1 x + tan^-1 z`
`= tan^-1 ((x+z)/(1-xz))`
`tan(2 tan^-1 y) = (x +z)/(1-xz)`
`2 tan(tan-1 y)/(1- (tan (tan^-1 y))^2 = (x+z)/(1-xz)`
`(2y)/(1-y^2) = (x+z)/(1- xz)`
`(x+z)/(1 - y^2) = (x+z)/(1 - xz)`
`(x+z)(1/(1-y^2) - 1/(1-xz)) = 0`
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