If   ${\mathrm{x}}^2+{\mathrm{y}}^2=\mathrm{t}+\frac{1}{t}$  and   ${\mathrm{x}}^4+{\mathrm{y}}^4={\mathrm{t}}^2+\frac{1}{{\mathrm{t}}^2}$  then  $\frac{dy}{dx}=$

${\text{x}}^{\text{2}}{\text{\hspace{0.17em}+\hspace{0.17em}y}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}t\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{\text{t}}$

${\left({\text{x}}^{\text{2}}{\text{\hspace{0.17em}+\hspace{0.17em}y}}^{\text{2}}\right)}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}}{\left(\text{t\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{\text{t}}\right)}^{\text{2}}$

${\text{x}}^{\text{4}}{\text{\hspace{0.17em}+\hspace{0.17em}y}}^{\text{4}}{\text{\hspace{0.17em}+\hspace{0.17em}2x}}^{\text{2}}{\text{\hspace{0.17em}y}}^{\text{2}}{\text{\hspace{0.17em}=\hspace{0.17em}t}}^{\text{2}}\text{\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{{\text{t}}^{\text{2}}}\text{\hspace{0.17em}+\hspace{0.17em}2}$

${\text{t}}^{\text{2}}\text{\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{{\text{t}}^{\text{2}}}{\text{\hspace{0.17em}+\hspace{0.17em}2x}}^{\text{2}}{\text{\hspace{0.17em}y}}^{\text{2}}{\text{\hspace{0.17em}=\hspace{0.17em}t}}^{\text{2}}\text{\hspace{0.17em}+\hspace{0.17em}}\frac{\text{1}}{{\text{t}}^{\text{2}}}\text{\hspace{0.17em}+\hspace{0.17em}2}$

$\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2x}}^{\text{2}}{\text{\hspace{0.17em}y}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}2}$

$\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\hspace{0.17em}y\hspace{0.17em}=\hspace{0.17em}}1$

Diff wrt x

$\text{x\hspace{0.17em}}\frac{\text{dy}}{\text{dx}}\text{+y\hspace{0.17em}=0}$

$\Rightarrow \frac{\text{dy}}{\text{dx}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{-y}}{\text{x}}$