If ( 9 ), x, (7 18 ) are in arithmetic progression and ( 9 ), y, … - Vidyadip

MCQ

If $\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)$ are also in arithmetic progression, then $|x-2 y|$ is equal to:

✓
$0$
B
$3$
C
$4$
D
$1$

✓

Answer

Correct option: A.

$0$

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

and $2 y=\tan \frac{\pi}{9}+\tan \frac{5 \pi}{18}$

so, $x-2 y=\frac{1}{2}\left(\tan \frac{\pi}{9}+\tan \frac{7 \pi}{18}\right)$

$-\left(\tan \frac{\pi}{9}+\tan \frac{5 \pi}{18}\right)$

$\Rightarrow|x-2 y|=\left|\frac{\cot \frac{\pi}{9}-\tan \frac{\pi}{9}}{2}-\tan \frac{5 \pi}{18}\right|$

$=\left|\cot \frac{2 \pi}{9}-\cot \frac{2 \pi}{9}\right|=0$

$\left(\operatorname{as\,\,\,\,tan} \frac{5 \pi}{18}=\cot \frac{2 \pi}{9} ; \tan \frac{7 \pi}{18}=\cot \frac{\pi}{9}\right)$

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