JEE Main2013MathematicsSequences and SeriesActual
If x , y , z are positive numbers in A . P . and tan - 1 x , tan - 1 y and tan - 1 z are also in A . P . , then which of the following is correct.
Options
- A6 x = 3 y = 2 z
- B6 x = 4 y = 3 z
- Cx = y = z
- D2 x = 3 y = 6 z
Correct answer
C. x = y = z
Step-by-step solution
Given, tan - 1 x ,   tan - 1 y and tan - 1 z are in A . P . , so 2 tan - 1 y = tan - 1 x + tan - 1 z ⇒ tan - 1 y - tan - 1 x = tan - 1 z - tan - 1 y y − x 1 + x y = z − y 1 + y z     . . . i Given x ,   y ,   z are in A . P . i.e., 2 y = x + z ⇒ y − x = z − y Hence, from i 1 + x y = 1 + y z x y = y z either y = 0 (rejected) or x = z Hence, x   =   y   =   z .