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The general solution of the differential equation
[JEE Main-2020 (September)]
\( \mathrm{P} \)
\( \sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \frac{d y}{d x}=0 \) is :
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(where \( \mathrm{C} \) is a constant of integration)
(a) \( \sqrt{1+\mathrm{y}^{2}}+\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}+1}{\sqrt{1+\mathrm{x}^{2}}-1}\right)+\mathrm{C} \)
(b) \( \sqrt{1+\mathrm{y}^{2}}+\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}-1}{\sqrt{1+\mathrm{x}^{2}}+1}\right)+\mathrm{C} \)
(c) \( \sqrt{1+y^{2}}-\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}-1}{\sqrt{1+x^{2}}+1}\right)+C \)
IDves (d) \( \sqrt{1+\mathrm{y}^{2}}-\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}+1}{\sqrt{1+\mathrm{x}^{2}}-1}\right)+\mathrm{C} \)
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