A) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\]
B) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+x\frac{dy}{dx}=0\]
C) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}=0\]
D) \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\]
Correct Answer: C
Solution :
[c] Any conic whose axes coincide with coordinate axis is \[a{{x}^{2}}+b{{y}^{2}}=1\] (i) Diff. both sides w.r.t. \['x'\], we get \[2ax+2by\frac{dy}{dx}=0\] i.e., \[ax+by\frac{dy}{dx}=0\](ii) Diff. again, \[a+b\left( y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}} \right)=0\](iii) From (ii), \[\frac{a}{b}=-\frac{ydy/dx}{x}\] From (iii), \[\frac{a}{b}=-\left( y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}} \right)\] \[\therefore \frac{y\frac{dy}{dx}}{x}=y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}\] \[\Rightarrow xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}=0\]
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