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)

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

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