A) \[\frac{2xz}{x+z}\]
B) \[\frac{xz}{2(x-z)}\]
C) \[\frac{xz}{2(z-x)}\]
D) \[\frac{2xz}{(x-z)}\]
Correct Answer: A
Solution :
We have, \[{{a}^{x}}={{b}^{y}}={{c}^{z}}\] and \[{{b}^{2}}=ac\] For \[{{a}^{x}}={{b}^{y}}\Rightarrow {{a}^{x/y}}={{b}^{y/y}}\] \[\Rightarrow b={{a}^{x/y}}\] ?(i) Also, \[{{c}^{z}}={{a}^{x}}\Rightarrow {{c}^{z/z}}={{a}^{x/z}}\] \[\Rightarrow c={{a}^{x/z}}\] ?(ii) Now, \[{{b}^{2}}=ac\] \[\Rightarrow {{({{a}^{x/y}})}^{2}}=a\times {{a}^{x/z}}\] [From (i) and (ii)] \[\Rightarrow {{a}^{2x/y}}={{a}^{((x/z)+1)}}\] On comparing, we get \[\frac{2x}{y}=\frac{x}{z}+1\] \[\Rightarrow \frac{2x}{y}=\frac{x+z}{z}\Rightarrow y=\frac{2xz}{x+z}\]
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