If $ 2(y-a) $ is the harmonic mean between $ y-x $ and $ y-z $, then $ x-a, y-a $ and $ z-a $ are in

Question

If $ 2(y-a) $ is the harmonic mean between $ y-x $ and $ y-z $, then $ x-a, y-a $ and $ z-a $ are in, arithmetic progression, geometric progression, harmonic progression, none of these

MARKS App · Accepted Answer

It is given that, y - x , 2 y - a a n d y - z are in harmonic progression. ⇒ 1 y - x , 1 2 y - a , 1 y - z are in arithmetic progression. ⇒ 1 2 y - a - 1 y - x = 1 y - z - 1 2 y - a ⇒ 2 a - y - x y - x = y + z - 2 a y - z ⇒ x - a + y - a x - a - y - a = y - a + z - a y - a - z - a ⇒ x - a y - a = y - a z - a Hence, x - a , y - a a n d z - a are in geometric progression

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