If x, y, z are in A.P. and ^-1x, ^-1y and ^-1z are also in A.P., then: - Mathematics MCQ JEE Main 2026 | ExamTest.live

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If x, y, z are in A.P. and tan^{- 1} x, tan^{- 1} y and tan^{- 1} z are also in A.P., then:

Solution & Step-by-step Explanation

1. x, y, z in A.P. \Rightarrow 2 y = x + z .2. tan^{- 1} x, tan^{- 1} y, tan^{- 1} z in A.P. \Rightarrow 2 tan^{- 1} y = tan^{- 1} x + tan^{- 1} z .3. tan^{- 1} (\frac{2 y}{1 - y ^{2}}) = tan^{- 1} (\frac{x + z}{1 - x z}) .4. \frac{2 y}{1 - y ^{2}} = \frac{x + z}{1 - x z} .5. Substitute x + z = 2 y : \frac{2 y}{1 - y ^{2}} = \frac{2 y}{1 - x z} \Rightarrow 1 - y^{2} = 1 - x z \Rightarrow y^{2} = x z .6. Since 2 y = x + z and y^{2} = x z, x, y, z are both in A.P. and G.P., which implies x = y = z .

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If x, y, z are in A.P. and tan^{- 1} x, tan^{- 1} y and tan^{- 1} z are also in A.P., then:

2x = 3y = 6z

6x = 3y = 2z

6x = 4y = 3z

x = y = z

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