Question

If $x,y,z$ are non zero numbers in A.P. and ${\tan }^{-1}x,{\tan }^{-1}y,{\tan }^{-1}z$ are also in A.P., then

• A. $x=y=z$
• B. $2x=3y=6z$
• C. $6x=3y=2z$
• D. $6x=4y=3z$

Answer 1

The correct option is A $x=y=z$

$x,y,z$ are in A.P., so
$2y=x+z$
As ${\tan }^{-1}x,{\tan }^{-1}y,{\tan }^{-1}z$ are also in A.P., so
$2{\tan }^{-1}y={\tan }^{-1}x+{\tan }^{-1}z\cdots \left(1\right)$
Taking $\tan$ on both sides,
$\frac{2y}{1-y^2}=\frac{x+z}{1-xz}\Rightarrow 2y(y^2-xz)=0\Rightarrow y^2=xz(\because y\ne 0)$
$\therefore x,y,x$ are in A.P. and G.P. both, so $x=y=z$
Checking for $x=y=z$ in equation $\left(1\right)$
It satisfies, hence $x=y=z$