QJEEM21Ma9903: JEE Main 2021 Mathematics - Class 11 JEEM21Ma9903 Solution: If are in arithmetic progression and are also in arithmetic…

Question

Verified Solution • 2025–26 Session moderate

If

\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)

are in arithmetic progression and

\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)

are also in arithmetic progression, then

|x - 2y|

is equal to :

Accepted Answer

x = {1 \over 2}\left( {\tan {\pi \over 9} + \tan {{7\pi } \over {18}}} \right)

and

2y = \tan {\pi \over 9} + \tan {{5\pi } \over {18}}

If we interpret the angles in degrees (as suggested by the numbers 20, 50, and 70), we have :

x = \frac{1}{2} \left( \tan 20^\circ + \tan 70^\circ \right),

and

2y = \tan 20^\circ + \tan 50^\circ.

The expression for

|x - 2y|

is then :

|x - 2y| = \left|\frac{\tan 20^\circ + \tan 70^\circ}{2} - \left( \tan 20^\circ + \tan 50^\circ \right)\right|.

|x - 2y| = \left|\frac{\tan 20^\circ + \tan 70^\circ - 2 \tan 20^\circ - 2 \tan 50^\circ}{2}\right|,

which simplifies to :

|x - 2y| = \left|\frac{\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ}{2}\right|.

We know,

\begin{aligned}&amp; \tan 70=\frac{\tan 20+\tan 50}{1-\tan 20 \tan 50} \\\\&amp;\Rightarrow \tan 70-\tan 70 \cdot \tan 20 \tan 50=\tan 20+\tan 50 \\\\&amp;\Rightarrow \tan 70-\tan 50-\tan 20-\tan 50=0 \\\\&amp;\Rightarrow \tan 70-\tan 20-2 \tan 50=0\end{aligned}

\therefore

|x - 2y| = \left|\frac{\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ}{2}\right|

=

\left| {{0 \over 2}} \right|

= 0

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