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(A) The $4^{th}$ roots of unity are $1, -1, i, -i$. The real roots are $\alpha=1, \beta=-1$ and the other roots are $\gamma=i, \delta=-i$.For the firs

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$\frac{5}{6}$

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(A) The $4^{th}$ roots of unity are $1, -1, i, -i$. The real roots are $\alpha=1, \beta=-1$ and the other roots are $\gamma=i, \delta=-i$.For the first conic $|z-1|+|z+1|=4$,this represents an ellipse with foci at $(\pm 1, 0)$ and major axis $2a=4$,so $a=2$. The distance between foci is $2ae_1=2$,so $e_1=\frac{1}{2}$.For the second conic $|z-i|+|z+i|=6$,this represents an ellipse with foci at $(0, \pm 1)$ and major axis $2a=6$,so $a=3$. The distance between foci is $2ae_2=2$,so $e_2=\frac{1}{3}$.The sum of the eccentricities is $e_1+e_2 = \frac{1}{2}+\frac{1}{3} = \frac{5}{6}$.

If $\alpha, \beta$ are the two real roots of the $4^{th}$ roots of unity and $\gamma, \delta$ are the other two roots,then the sum of the eccentricities of the conics $|z-\alpha|+|z-\beta|=4$ and $|z-\gamma|+|z-\delta|=6$ is

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