• # question_answer The differential equations of all conies whose axes coincide with the co-ordinate axis 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) $xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0$

[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), $\frac{a}{b}=-\left( y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}} \right)$ $\therefore \frac{y\frac{dy}{dx}}{x}=y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}$ $\Rightarrow xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}=0$