7. A line segment connecting two different points on an ellipse is called a chord of the ellipse. Then, the equation of the locus of the midpoint of a chord of length $\sqrt{5}$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ is $\qquad$
7. $x^{4}+20 x^{2} y^{2}+64 y^{4}+x^{2}-44 y^{2}=0$.

Let the endpoints of the chord be
$$
A(2 \cos \theta, \sin \theta), B(2 \cos \varphi, \sin \varphi),
$$

and the midpoint of segment $A B$ be $C\left(x_{0}, y_{0}\right)$.
Then $\left\{\begin{array}{l}x_{0}=2 \cos \frac{\theta+\varphi}{2} \cdot \cos \frac{\theta-\varphi}{2}, \\ y_{0}=\sin \frac{\theta+\varphi}{2} \cdot \cos \frac{\theta-\varphi}{2} .\end{array}\right.$
Dividing the two equations gives $\tan \frac{\theta+\varphi}{2}=\frac{2 y_{0}}{x_{0}}$.
Using the universal formula, we get
$$
\cos (\theta+\varphi)=\frac{1-\tan ^{2} \frac{\theta+\varphi}{2}}{1+\tan ^{2} \frac{\theta+\varphi}{2}}=\frac{x_{0}^{2}-4 y_{0}^{2}}{x_{0}^{2}+4 y_{0}^{2}} .
$$

Squaring and adding the two equations in the system, we get
$$
\begin{array}{l}
x_{0}^{2}+4 y_{0}^{2}=2+2 \cos (\theta-\varphi) \\
\Rightarrow \cos (\theta-\varphi)=\frac{1}{2}\left(x_{0}^{2}+4 y_{0}^{2}\right)-1 \text {. } \\
\text { Hence }\left\{\begin{array}{l}
\cos \theta \cdot \cos \varphi=\frac{1}{4}\left(x_{0}^{2}+4 y_{0}^{2}\right)-\frac{1}{2}+\frac{x_{0}^{2}-4 y_{0}^{2}}{2\left(x_{0}^{2}+4 y_{0}^{2}\right),} \\
\sin \theta \cdot \sin \varphi=\frac{1}{4}\left(x_{0}^{2}+4 y_{0}^{2}\right)-\frac{1}{2}-\frac{x_{0}^{2}-4 y_{0}^{2}}{2\left(x_{0}^{2}+4 y_{0}^{2}\right)} \text {. }
\end{array}\right. \\
\end{array}
$$
$$
\text { Then } \begin{array}{l} 
5=(2 \cos \theta-2 \cos \varphi)^{2}+(\sin \theta-\sin \varphi)^{2} \\
= 4(\cos \theta+\cos \varphi)^{2}-16 \cos \theta \cdot \cos \varphi+ \\
(\sin \theta+\sin \varphi)^{2}-4 \sin \theta \cdot \sin \varphi \\
=-x_{0}^{2}-16 y_{0}^{2}+10-\frac{6\left(x_{0}^{2}-4 y_{0}^{2}\right)}{x_{0}^{2}+4 y_{0}^{2}} .
\end{array}
$$

Therefore, $\left(x_{0}, y_{0}\right)$ satisfies the equation
$$
x^{4}+20 x^{2} y^{2}+64 y^{4}+x^{2}-44 y^{2}=0 \text {. }
$$