A) \[2({{x}^{2}}-{{y}^{2}})y=xy\]
B) \[2({{x}^{2}}+{{y}^{2}})y=xy\]
C) \[({{x}^{2}}-{{y}^{2}})y=2xy\]
D) \[({{x}^{2}}+{{y}^{2}})y=2xy\]
Correct Answer: C
Solution :
The equation of the family of curves is \[{{x}^{2}}+{{y}^{2}}-2ay=0\] ...(i) On differentiating w.r. to\[x\] \[2x+2yy-2ay=0\] \[\Rightarrow \] \[2x+2yy=2ay\] \[\Rightarrow \] \[\frac{2x+2yy}{y}=2a\] ?(ii) From Eq. (i) \[2a=\frac{{{x}^{2}}+{{y}^{2}}}{y}\] On putting this value in Eq. (ii) \[\frac{2x+2yy}{y}=\frac{{{x}^{2}}+{{y}^{2}}}{y}\] \[\Rightarrow \] \[2xy+2{{y}^{2}}y={{x}^{2}}y+{{y}^{2}}y\] \[\Rightarrow \] \[({{x}^{2}}-{{y}^{2}})y=2xy\]
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