If \( \Large a^{x} = b^{y} = c{z} \) and \( \Large \frac{b}{a} = \frac{c}{b} \) then \( \Large \frac{2z}{x + z} = ? \)


A) \( \Large \frac{y}{x} \)

B) \( \Large \frac{x}{y} \)

C) \( \Large \frac{x}{z} \)

D) \( \Large \frac{z}{x} \)

Correct Answer:
A) \( \Large \frac{y}{x} \)

Description for Correct answer:
\( \Large \frac{b}{a} = \frac{c}{a} => b^{2} = ca \)

Again, \( \Large a^{x} = b^{y} = c^{z} = k \)

=> \( \Large a = k^{\frac{1}{x}}, b = k^{\frac{1}{y}} , c = k^{\frac{1}{z}} \)

\( \Large \therefore b^{2} = ac \)

=> \( \Large k ^{\frac{2}{y}} = k^{\frac{1}{x}} . k^{\frac{1}{z}} \)

=> \( \Large \frac{2}{y} = \frac{1}{x} + \frac{1}{z} = \frac{z + x}{zx} \)

=> \( \Large \frac{2z}{x + z} = \frac{y}{x} \)

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