Discussion

If \(a^x = b^y = c^z\) and \(b^2 = ac\), then \(y\) equals:

Answer: D

Let \(a^x = b^y = c^z = k\).

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

Therefore, \(b^2 = ac \)

\(\Rightarrow (k^{\frac{1}{y}})^2 = k^{\frac{1}{x}} \times k^{\frac{1}{z}} \)

\(\Rightarrow k^{\frac{2}{y}} = k^{(\frac{1}{x} + \frac{1}{z})}\)

Therefore, \(\frac{2}{y} = \frac{(x+z)}{xz} \)

\(\Rightarrow \frac{y}{2} = \frac{xz}{(x+z)} \)

\(\Rightarrow y = \frac{2xz}{(x + z)}\)

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