Algebra

If 2(y-a) is HM between y-x and y-z then x-a, y-a, I-a are in
Ap
Hp
Gp
None

Anvita Mahajan

7 Years agoGrade

7 Years ago

2(y – a) is HM between y – x and y – z. $\Rightarrow$ y – x , 2(y – a) , y – z are in HP.

i.e. 1 / (y – x) , 1 / 2(y – a) , 1 / (y – z) are in AP.

$\therefore$ 2 . {1 / 2(y – a)} = 1 / (y – x) + 1 / (y – z)

1 / (y – a) = 1 / (y – x) + 1 / (y – z) $\Rightarrow$ 1 / (y – a) – 1 / (y – x) = 1 / (y – z)

(y – x – y + a) / (y – a)(y – x) = 1 / (y – z)

– (x – a) / (y – a)(y – x) = 1 / (y – z) $\therefore$ (x – a) / (y – x) = (y – a) / (z – y)

(x – a) / [(y – a) – (x – a)] = (y – a) / [(z – a) – (y – a)]

By invertendo, [(y – a) – (x – a)] / (x – a) = [(z – a) – (y – a)] / (y – a)

i.e. {(y – a) / (x – a)} – 1 = {(z – a) / (y – a)} – 1 $\Rightarrow$ (y – a) / (x – a) = (z – a) / (y – a)

$\therefore$ (y – a)² = (x – a)(z – a) i.e. (x – a) , (y – a) , (z – a) are in GP.

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