Question 110, Given that the set of points in the complex plane corresponding to the complex number $z$ satisfying the condition $\left|z^{2}\right|+\left|z^{2}-1\right|=7$ forms a quadratic curve, find the eccentricity $e$ of this quadratic curve.

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Question 110, Given that the set of points in the complex plane corresponding to the complex number $z$ satisfying the condition $\left|z^{2}\right|+\left|z^{2}-1\right|=7$ forms a quadratic curve, find the eccentricity $e$ of this quadratic curve.
Let $\mathrm{z}=\mathrm{x}+\mathrm{yi}$, then
$$
\begin{array}{l}
\left|(x+y i)^{2}\right|+\left|(x+y i)^{2}-1\right|=7 \\
\Rightarrow x^{2}+y^{2}+\sqrt{\left(x^{2}-y^{2}-1\right)^{2}+(2 x y)^{2}}=7 \\
\Rightarrow\left[7-\left(x^{2}+y^{2}\right)\right]^{2}=\left(x^{2}-y^{2}-1\right)^{2}+(2 x y)^{2}=\left(x^{2}+y^{2}\right)^{2}-2\left(x^{2}-y^{2}\right)+1 \\
\Rightarrow 12 x^{2}+16 y^{2}=48 \\
\Rightarrow \frac{x^{2}}{4}+\frac{y^{2}}{3}=1
\end{array}
$$

Thus, $a=2, b=\sqrt{3}, c=1$, so $e=\frac{c}{a}=\frac{1}{2}$.