Question

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

• A. $xy=yz$
• B. $z^2=xy$
• C. $x=y=z$
• D. $x^2=yz$

Answer 1

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

We have
$x,y,z$ are in AP
$\Rightarrow 2y=x+z...\left(i\right)$
Also,
$\Rightarrow 2{\tan }^{-1}y={\tan }^{-1}z+{\tan }^{-1}x$
$\Rightarrow {\tan }^{-1}\left(\frac{2y}{1-y^2}\right)={\tan }^{-1}\left(\frac{x+z}{1-xz}\right)$
$\Rightarrow \frac{2y}{1-y^2}=\frac{x+z}{1-xz}$
$\Rightarrow 1-y^2=1-xz$ (Using Eq (i))
$y^2=xz...\left(ii\right)$
$\Rightarrow$ $x,y,z$ are in G.P
Using Eqs. (i) and (ii) we get $x=y=z$