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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:

Option 1)
$6x=4y=3z$

Option 2)
$x=y=z$

Option 3)
$2x=3y=6z$

Option 4)
$6x=3y=2z$

Answers (1)

best_answer

As we learnt in

Double Angle Formula -

Double angle formula

- wherein

These are formulae for double angles.

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

$\Rightarrow 2tan^{-1}y=tan^{-1}x+tan^{-1}z$ $= tan^{-1} = \frac{x+z}{1-xz}$

$\Rightarrow tan^{-1} = \frac{2y}{1-y^2}= \frac{x+z}{1-xz}$

$\Rightarrow {\frac{2y}{1-y^2}}=\frac{x+z}{1-xz}$

$\Rightarrow 1-y^2= 1-xz\left [ as\ y = \frac{x+z}{2} \right ]$

$\Rightarrow xz=y^2............\left ( 1 \right )$

But $y= \frac{x+z}{2}..................\left ( 2 \right )$

from (1) and (2) $\left ( \frac{x+z}{2} \right )^2=xz$

$\Rightarrow \left ( x-z \right )^2=0$

Thus $x=y=z$

Option 1)

$6x=4y=3z$

Incorrect

Option 2)

$x=y=z$

Correct

Option 3)

$2x=3y=6z$

Incorrect

Option 4)

$6x=3y=2z$

Incorrect

Posted by

prateek

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