A) \[y:x\]
B) \[x:y\]
C) \[-y:x\]
D) \[-x:y\]
Correct Answer: C
Solution :
\[\frac{\partial z}{\partial x}=\frac{1}{1+\frac{{{x}^{2}}}{{{y}^{2}}}}.\frac{1}{y}=\frac{y}{{{x}^{2}}+{{y}^{2}}}\] \[\frac{\partial z}{\partial y}=\frac{1}{1+\frac{{{x}^{2}}}{{{y}^{2}}}}.\left( -\frac{x}{{{y}^{2}}} \right)=-\frac{x}{{{x}^{2}}+{{y}^{2}}}\] \[\therefore \] \[\frac{\partial z}{\partial x}:\frac{\partial z}{\partial y}=y:-x\] i.e., \[{{z}_{x}}:{{z}_{y}}=-y:x\]
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