$x, y, z$ are in G.P. and $\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z$ are in A.P., then

$x, y, z$ are in G.P. and $\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z$ are in A.P., then
  1. $6 x=4 y=3 z$
  2. $2 x=3 y=6 z$
  3. $6 x=3 y=2 z$
  4. $x=y=z$

Solution

$x, y, \mathrm{z}$ are in G.P. $\Rightarrow y^2=x \mathrm{z}$ Also, $\tan ^{-1} x, \tan ^{-1} y, \tan ^{-1} z$ are in A.P. $\begin{aligned} & \Rightarrow 2 \tan ^{-1} y=\tan ^{-1} x+\tan ^{-1} \mathrm{z} \\ & \Rightarrow \tan ^{-1}\left(\frac{2 y}{1-y^2}\right)=\tan ^{-1}\left(\frac{x+\mathrm{z}}{1-x \mathrm{z}}\right) \\ & \Rightarrow \frac{2 y}{1-y^2}=\frac{x+\mathrm{z}}{1-x \mathrm{z}} \\ & \Rightarrow \frac{2 y}{1-x \mathrm{z}}=\frac{x+\mathrm{z}}{1-x \mathrm{z}} \end{aligned}$ $\Rightarrow 2 y=x+\mathrm{z}$ $\Rightarrow x, y, \mathrm{z}$ are in A.P. From (i) and (ii), we get $x=y=\mathrm{Z}$

Asked in: MHT CET 2023 (13 May Shift 1)

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