A) \[x=y=z\]
B) \[2x=3y=6z\]
C) \[6x=3y=2z\]
D) \[6x=4y=3z\]
Correct Answer: A
Solution :
\[2y=x+z\] ? (1) As \[ta{{n}^{-1}}x,\text{ }ta{{n}^{-1}}y,\text{ }ta{{n}^{-1}}z\] in AP \[\Rightarrow \]\[2\text{ }ta{{n}^{-1}}y=ta{{n}^{-1}}\frac{x+z}{1-xz}\] \[\frac{2y}{1-{{y}^{2}}}=\frac{x+z}{1-xz}\] \[\frac{x+z}{1-{{y}^{2}}}=\frac{x+z}{1-xz}\] ? by (1) \[(x+z)\left\{ \frac{1}{1-{{y}^{2}}}-\frac{1}{1-xz} \right\}=0\] \[x+z=0\text{ }or\text{ }1-xz=x-{{y}^{2}}\] \[{{y}^{2}}=xz\] \[\Rightarrow \] \[x,\text{ }y,\text{ }z\] in GP. As \[x,\text{ }y,\text{ }z\] AP & GP \[\Rightarrow \] \[x=y=z\]
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